There is such a rule, but the rule depends on context:

It is not strictly correct that between is used for two things and among for more than two.

When exactly two entities are specified, between should always be used: “This contract is entered into between the Seller and the Purchaser.”

However, when more than two entities are involved or when the number of entities is unspecified, the word choice depends on what you want to say. Between should be used where the relationship is distinctly one-to-one:

“The agreement was entered into between the Seller, the Purchaser and the Guarantor.”

Among should be used where the entities are considered as a group, mass or collectivity:

“There is consensus among shareholders that this approach be adopted.”

In your sentence, I would definitely have used between, as the relationship is clearly one-to-one. It would make more sense too.

The general rule as I've always learned is the same one you mention, but maybe it's a bit imprecise and it can lead to the misunderstanding you've highlighted in your question.

A more precise rule, supported by the quote below, but that you can see in other answers as well, is that between is used for distinct entities (people, things) and can actually be 2, 3 or more. Among indicates groups of people or objects, etc, that are not clearly "separated", highlighted in a specific manner or, indeed, distinct.

So your sentence works:

Between Cook, Strauss, and Pietersen, they’ve notched up 56 centuries in Test cricket.

But instead you'd use among to say something like:

I feel comfortable among my friends.

This is the quote I mentioned above (from the NOAD), check the third paragraph (double line):

USAGE
Between is used in speaking of only two things, people, etc.: "we must choose between two equally unattractive alternatives."

Among is used for collective and undefined relations of usually three or more: "Agreement on landscaping was reached among all the neighbors."

But where there are more than two parties involved, between may be used to express one-to-one relationships of pairs within the group or the sense 'shared by': "There is close friendship between the members of the club."; diplomatic relations between the U.S., Canada, and Mexico..

Between you and I,: between you and he, etc., are incorrect; between should be followed only by the objective case: between you and me,: between you and him, etc.

I like Strunk and White's simple example, from The Elements of Style:

When more than two things or persons are involved, among is usually called for: "The money was divided among the four players." When, however, more than two are involved but each is considered individually, between is preferred: "an agreement between the six heirs."

Prescriptivist Poppycock

You should in this instance trust your own ear, not your friend’s opinion that there is a rule that says you are wrong. There isn’t, and you aren’t. Between has never been restricted to two items alone. In the citations below, this non-rule your friend alludes to is variously called a persistent but unfounded notion, a superstition slow to die, and prescriptivist poppycock.

---

A Rule That Doesn’t Rule

I recommend William Safire’s “On Language” column entitled “Betwixt Among And Between”. Safire answers — and at length — a correspondent who encountered much the same quandary as you did. He rightly calls the bogus “rule” that your friend sought to impose on you “a rule that doesn’t really rule”; one that “hangs on with no real authority”. Safire then goes on to try to discover who was first responsible for this nonsense.

Where did the “rule” begin? Who laid this guilt on Ms. Thorp? “Between is properly used of two,” Samuel Johnson wrote in his dictionary of 1755, “and among of more.” He saw tween as meaning “twain, two,” and therefore decided that it would be “proper” to limit the division to two. He did notice, however, that not everybody was as fastidious as him (or, as he would say, as he): “But perhaps,” he added, “this accuracy is not always preserved.”

Whether it is the esteemed Dr Johnson or anyone else doing the judging, a word’s meaning or “correctness” can never be judged by studying what the original words that gave rise to the current word once meant. This is called the etymological fallacy. It is never sound reasoning: words mean only what they mean here and now, not what they might have meant somewhen else.

Later lexicographers took issue with Johnson’s statement. Fowler calls this notion that one must always use between for two and never more than that a superstition that dies hard. Not long after Johnson, Noah Webster wrote in his dictionary of 1828 that between is not restricted to two, and the current entry for between in Merriam-Webster further expands upon this rebuttal:

There is a persistent but unfounded notion that between can be used only of two items and that among must be used for more than two. Between has been used of more than two since Old English; it is especially appropriate to denote a one-to-one relationship, regardless of the number of items. It can be used when the number is unspecified 〈economic cooperation between nations〉, when more than two are enumerated 〈between you and me and the lamppost〉 〈partitioned between Austria, Prussia, and Russia — Nathaniel Benchley〉, and even when only one item is mentioned (but repetition is implied) 〈pausing between every sentence to rap the floor — George Eliot〉. Among is more appropriate where the emphasis is on distribution rather than individual relationships 〈discontent among the peasants〉. When among is automatically chosen for more than two, English idiom may be strained 〈a worthy book that nevertheless falls among many stools — John Simon〉 〈the author alternates among modern slang, clichés and quotes from literary giants — A. H. Johnston〉.

OED: “In all senses, between has been, from its earliest appearance, extended to more than two.”

Being an historical dictionary aware of English in all its incarnations, the OED has in every edition disagreed with the idea that between could not be used for more than two. One of its Middle English citations quite specifically writes Betweyn us thre — that is, between us three. And three is without question greater than two.

In the second edition of the OED, this statement is made:

V. 19. In all senses, between has been, from its earliest appearance, extended to more than two. In OE. and ME. it was so extended in sense 1, in which among is now considered better. It is still the only word available to express the relation of a thing to many surrounding things severally and individually, among expressing a relation to them collectively and vaguely: we should not say ‘the space lying among the three points,’ or ‘a treaty among three powers,’ or ‘the choice lies among the three candidates in the select list,’ or ‘to insert a needle among the closed petals of a flower.’

As any native speaker immediately realizes, those four examples which the OED says that we “should not say” are failures as phrases go.

In the various citations the OED provides to document historical uses of between applied to more than two items (the earliest is from 971), these two stand out most ironically, since they are (allegedly) by the same author contradicting himself:

• 1755 Johnson Dict., ― Between is properly used of two, and among of more; but perhaps this accuracy is not always preserved.
• 1771 Johnson in Boswell (1826) II. 127, ― I··hope, that, between publick business, improving studies, and domestick pleasures, neither melancholy nor caprice will find any place for entrance.

So if Boswell is correctly quoting Johnson, then Johnson seems to have failed to take his own advice regarding propriety and accuracy. Apparently the good doctor’s native ear got the better of him — and a good thing, too, for it would have sounded silly for him to use among there.

---

“When zombie rules attack!”

In his Language Log article “Learning to speak Imaginary American”, linguist Mark Liberman identifies Goold Brown’s 1851 Grammar of English Grammars as the progenitor of what Liberman has filed under “prescriptivist poppycock”:

The idea that between must be used for two alternatives, and among for more than two, is a Zombie Rule with a pedigree. It was apparently invented by Goold Brown in 1851, in his Grammar of English Grammars, in order to demonstrate his superiority to earlier grammarians who had “misused” between for more than two alternatives.

If this is the origin point of uncounted mindless repetitions of a bogus rule foisted upon billions of English speakers and learners, it is is illustrative to examine exactly what Brown said and how he said it.

or, if you would,

                                       The
                                     Grammar
                                       Of
                                English Grammars,
                                      With
                                 An Introduction
                            Historical and Critical ;
                                    The Whole
                  Methodically Arranged and Amply Illustrated ;
                                      With
        Forms of Correcting and of Pausing, Improprieties For Correction,
     Examples for Parsing, Questions for Examination, Exercises For Writing,
                     Observations For the Advanced Student,
           Decisions and Proofs for the Settlement of Disputed Points,
                       Occasional Strictures and Defences,
                An Exhibition of the Several Methods of Analysis,
                                       And
                          A Key to the Oral Exercises :
                               To Which Are Added
                                Four Appendixes,
               Pertaining Separately to the Four Parts of Grammar.

                                 By Goold Brown,

           Formerly Principal of an English and Classical Academy, New York ;
                   Author of the Institutes English Grammar,
                   The First Lines of English Grammar, etc.

                      “So let great authors have their due,
                    that Time, who is the author of authors,
                           be not deprived of his due,
                         which is, farther and farther,
                         to discover truth.” ―Lord Bacon

                                   New York :
                     Published By Samuel S. & William Wood,
                               № 261 Pearl Street,
                                      1851.

_{(Interesting quote from Lord Bacon, eh? Apparently the truth is something to be “farther” discovered. Hm...)}

Could the author of so humbly titled a work truly have attempted to show his “superiority” over previous grammarians, as Liberman writes?

Perhaps not, but let us examine the evidence and decide for ourselves.

Brown answers his own question of

What notice is taken of the application of between, betwixt, among, amongst, amid, amidst?

by observing on page 653 that

Obs. 13. ―Between, or betwixt, is used in in reference to two things or parties ; among, or amonst, amid, or amidst, in reference to a greater number, or to something by which an other may be surrounded : as, “Thou pendulum betwixt a smile and tear.” ―Byron “The host between the mountain and the shore.” ―Id. “To meditate amongst decay, and stand a ruin amidst ruins.” ―Id. In the following examples, the import of these prepositions is not very accurately regarded ; “Ihe Greeks wrote in capitals, and left no spaces between their words.” ―Wilson’s Essay, p. 6. This construction may perhaps be allowed, because the spaces by which words are now divided, occur severally between one word and an other ; but the author might as well have said, “and left no spaces to distinguish their words.” “There was a hunting match agreed upon betwixt a lion, an ass, and a fox.” ―L’Estrange. Here by or among would, I think, be better than betwixt, because the partners were more than two. “Between two or more authors, different readers will differ exceedingly, as to the preference in point of merit.” ―Campbell’s Rhet. p. 162; Jamieson's, 40 ; Murray’s Gram., i, 360. Say, “Concerning two or more authors,” because between is not consistent with the word more. “Rising one among another in the greatest confusion and disorder.” ―Spect. No. 476. Say, “Rising promiscuously,” or, “Rising all at once ; ” for among is not consistent with the distributive term one an other.

Even Brown himself can bring himself only to say that he “thinks” such-and-such would be better. There is no such rule. This is just a peever spouting off his pet peeve in a published format.

The problem is that people mistook a pet peeve for an actual rule of English grammar, like plural subjects taking plural verbs. It might be a usage recommendation (whether sound or otherwise), but a rule of grammar it is not.

But it gets worse. Brown digs his hole still deeper.

On page 657 Brown lays out more than a dozen “mistakes” committed by earlier scholars, and then proceeds on page 948 to “correct” these defects. Collating the originals from page 657 with Brown’s corrigenda from page 948 immediately following each, these are as follows (with all italics exactly as show in Brown, not editorial ones adds by this author):

1. “The Anglo-Saxons, however, soon quarrelled between themselves for precedence.” ―Constable’s Miscellany, xx, p. 59.
   
   “The Anglo-Saxons, however, soon quarrelled among themselves for precedence.” ―Const. Miso. cor.
   
   
2. “The distinctions between the principal parts of speech are founded in nature.” ―Webster’s Essays, p. 7.
   
   “The distinctions among the principal parts of speech are founded in nature.” ―Webster cor.
   
   
3. “I think I now understand the difference between the active, passive, and neuter verbs.” ―Ingersoll’s Gram. p. 124.
   
   “I think I now understand the difference between the active verbs and those which are passive or neuter.” ―Ingersoll cor.
   
   
4. “Thus a figure including a space between three lines, is the real as well as nominal essence of a triangle.” ―Locke’s Essay, p. 303.
   
   “Thus a figure including a space within three lines, is the real as well as nominal essence of a triangle.” ―Locke cor.
   
   
5. “We must distinguish between an imperfect phrase, a simple sentence, and a compound sentence." ―Lowth’s Gram. p. 117 ; Murray’s, i, 267 ; Ingersoll’s, 280 ; Guy's, 97.
   
   “We must distinguish between an imperfect phrase and a simple sentence, and between a simple sentence and a compound sentence.” ―Lowth, Murray, et al. cor.
   
   
6. “The Jews are strictly forbidden by their law, to exercise usury among one an other.” ―Sale’s Koran, p. 177.
   
   “The Jews are strictly forbidden by their law, to exercise usury towards one an other.” ―Sale cor.
   
   
7. “All the writers have distinguished themselves among one another.” ―Addison.
   
   “All the writers have distinguished themselves among themselves.” ―Addison cor.
   
   
8. “This expression also better secures the systematic uniformity between the three cases.” ―Nutting’s Gram. p. 98.
   
   “This expression also better secures the systematic uniformity of the three cases.” ―Nutting cor.
   
   
9. “When a disjunctive occurs between two or more Infinitive Modes, or clauses, the verb must be singular.’ ―Jaudon’s Gram. p. 95.
   
   “When two or more infinitives or clauses are connected disjunctively as the subjects of an affirmation, the verb must be singular.” ―Jaudon cor.
   
   
10. “Several nouns or pronouns together in the same case, not united by and, require a comma between each.” ―Blair’s Gram. p. 115.
    
    “Several nouns or pronouns together in the same case, require a comma after each ; [except the last, which must sometimes be followed by a greater point.]” ―D. Blair cor.
    
    
11. “The difference between the several vowels is produced by opening the mouth differently, and placing the tongue in a different manner for each.” ―Churchill’s Gram. p. 2.
    
    “The difference between one vowel and an other is produced by opening the mouth differently, and placing the tongue in a different manner for each.” ―Churchill cor.
    
    
12. “Thus feet composed of syllables, being pronounced with a sensible interval between each, make a more lively impression than can be made by a continued sound.” ―Kames, El. of Grit. Vol. ii, p. 32.
    
    “Thus feet composed of syllables, being pronounced with a sensible interval between one foot and an other, make a more lively impression than can be made by a continued sound.” ―Kames cor.
    
    
13. “The superlative degree implies a comparison between three or more.” ―Smith’s Productive Gram. p. 51.
    
    "The superlative degree implies a comparison, sometimes between two, but generally among three or more.” ―Smith cor.
    
    
14. “They are used to mark a distinction between several objects.” ―Levizac’s Gram. p. 85.
    
    “They are used to mark a distinction among several objects.” ―Levizac cor.
    
    

There you go. That’s what Brown thought people should be writing. Not only are almost all of Brown’s rewrites less felicitous than the original versions, many are cumbersome, artificial, and tedious. Some are laughably bad. In places his changes even alter the sentence’s very meaning, making it say something other than the original intended.

Liberman was right, for you can see Brown’s condescension bleeding through as he “corrects” the authors of earlier grammars, who apparently just weren’t as smart as Brown (thought that he himself) was.

---

Summary

There is no such “rule”, and there never was.

This is another artificial anti-English “rule” invented by the same set of scolding prescriptivists who took it upon themselves to forbid us from “splitting infinitives” and dangling prepositions off our sentences’ ends. It has no basis in actual English.

Sure, there are indeed places where between won’t comfortably fit, just as there are places where among will not. There are also places where either fits.

But this comfort-zone, or lack thereof, is determined by more than any simplistic yes–no question of whether exactly two parties are involved can ever hope to answer.

By "correcting" you, your friend was perpetuating a myth that stems from an oversimplification. Here is a usage note from Merriam-Webster:

There is a persistent but unfounded notion that ‘between’ can be used only of two items and that ‘among’ must be used for more than two. ‘Between’ has been used of more than two since Old English; it is especially appropriate to denote a one-to-one relationship, regardless of the number of items.

Wikipedia, in an entry on disputed words, says this:

The traditionalist view is that ‘between’ should only be used when there are only two objects for comparison; and ‘among’ or ‘amongst’ should be used for more than two objects. Most style guides and dictionaries do not support this advice, saying that ‘between’ can be used to refer to something that is in the time, space or interval that separates more than two items. M-W says that the idea that between can be used only of two items is "persistent but unfounded" and AHD4 calls it a "widely repeated but unjustified tradition". The OED says "In all senses, between has been, from its earliest appearance, extended to more than two". Chambers says "It is acceptable to use ‘between’ with reference to more than two people or things", although does state that ‘among’ may be more appropriate in some circumstances.

Frances Peck, in her article "Usage Myths", says that:

When three or more entities relate to each other individually and equally rather than collectively, or when the relationship is not general but reciprocal, ‘between’ is the correct choice. For example: A free trade agreement between (not among) Canada, Mexico, and the United States.

A blogger writes:

It is not strictly correct that ‘between’ is used for two things and among for more than two. When more than two entities are involved or when the number of entities is unspecified, the word choice depends on what you want to say. ‘Between’ should be used where the relationship is distinctly one-to-one: “The agreement was entered into between the Seller, the Purchaser and the Guarantor.” ‘Among’ should be used where the entities are considered as a group, mass or collectivity: “There is consensus among shareholders that this approach be adopted.”

A grammar column summarized, amongst other "debunked myths":

The rule that ‘between’ can only be used with two items, and ‘among’ for more than two, is specious. The real tendency of English is for ‘between’ when the connections are conceptualized as being between individuals, and ‘among’ when the connections are more vague and collective.

Espresso English states:

It is often taught that ‘between’ is used for 2 items and ‘among’ for 3 or more. But this is not completely accurate. The more accurate difference is this:

• ‘Between’ is used when naming distinct, individual items (can be 2, 3, or more)
• ‘Among’ is used when the items are part of a group, or are not specifically named (MUST be 3 or more)

Grammar expert Richard Norquist, in a "commonly confused words" column, says:

In general, ‘between’ applies to reciprocal arrangements (one member to another member), and ‘among’ applies to collective arrangements (with all members involved). However .. ‘between’ may apply to more than two members.

Norquist goes on to quote Pam Peters:

‘Between’ was formerly reserved for situations where just two things or people were being related – “shared between husband and wife” – and ‘among’ complemented it when there were three or more: “shared among the relatives.” The restriction on the use of between has certainly gone by the board, and Gowers declared it to be 'superstition' in Complete Plain Words (1954). It is not uncommon for between to be used in expressions referring to more than two groups or reference points, as in “a balance between deference, quotation and his own critical comment.”

A University of Arizona faculty member writes:

The common advice these days is that ‘between’ should be reserved for the comparison of just two entities, and if there are more than two, ‘among’ should be used. What is wrong with this advice? The common advice is wrong because it leads to an incorrect use of ‘among.’

Traditionally, it was believed that you should use between if you’re talking about two people or things:

• "I had to choose between the two jobs."

• "The complex has 270 units, divided between two interior plazas."

But use among (or amongst) if you’re referring to more than two people or things:

• "Gallery-goers can pick and choose among the DVDs and watch various moments."

• "The grant will be divided among all the institutions."

Such advice is now regarded as outdated and out of step with current usage.

In modern English, it’s perfectly acceptable to use between or among in certain contexts when referring to more than two participants.

Here’s how that works in practice: Always use between, not among if you want to talk about a position within two specific points in time or on a scale:

• The children were aged between 12 and 16. (correct)✓

• The children were aged among 12 and 16. (incorrect)✖

Opt for between if you’re referring to any number of specific (i.e. named) people, groups, or things:

• A treaty was drawn up between France, Germany, Italy, and Greece. (correct)✓

• A treaty was drawn up among France, Germany, Italy, and Greece. (incorrect)✖

• Flights between London and Crete begin in early April. (correct)✓

• Flights among London and Crete begin in early April. (incorrect)✖

Use among if the people or things can be regarded collectively (that is, as a group or undifferentiated mass) rather than individually:

• There was agreement among members that fees should not be raised. (correct)✓

• There was agreement between members that fees should not be raised. (incorrect)✖

• The Petronas Towers are among the architectural wonders of the world. (correct)✓

• The Petronas Towers are between the architectural wonders of the world. (incorrect)✖

Between is preferred when we talk about a relationship of difference, no matter how many people or things are involved:

• The difference between those results is not statistically significant. (correct)✓

• The difference among those results is not statistically significant. (incorrect)✖

  1. Bill and Mark swam among the sunken ships.

  2. Bill and Mark swam between the sunken ships.

Both of these sentences are correct English, but they have different meanings. In the first example, as Bill and Mark were swimming, they were surrounded by several sunken ships. In the second example, they had a sunken ship (or ships) to the right of them and another (or others) to the left, and were swimming through the gap in the middle.