Independent clause markers

Words such as however, since they can do different things, are defined by their function more than their actual location in the sentence. Granted, position tends to dominate the role a word plays in a sentence but your best bet is to look at what job the word is doing

In your first example your marker word however is functioning as a conjunction. Thus the semicolon, comma, and clause as dictated by the protocol of conjunctions.

In the second example (which, you will note, does not need the two commas), your marker word is behaving like an adverb because it is a modifying the verb missed. No conjunction to be found, therefore no signal word. Use the grade-school standby questions on your alleged conjunctions to see if they're just being plain old adverbs.

• How?
• When?
• Where?
• In What Way?

This a handy trick since all it takes to change the function of words like still, consequently, and however is some clever rearrangement.

Adverbs will answer those questions and conjunctions will not. A word of caution, though: that last one in what way? is tricky, because conjunctive adverbs are still adverbs which means that sometimes a word like moreover will answer that question if it's combined with the right words.

The second example The bus was late. Many people consequently missed their appointments. contains two independent clauses, neither of which have a distinguishable marker. Independent clauses don't really need their own flag; all you need to test for an independent clause is a subject and verb (If identifying types of clauses is foggy, you should send me a private message or email).

To answer your question (finally, I apologize for my incessant verbosity):

Independent clauses can be marked by an adverb functioning as a conjunction Fun fact - adverbs that are capable of filling the shoes of a conjunction are called conjunctive adverbs (surprise!) but you can keep calling them "marker words" if that's what works for you.

Then boom. You bust out the semicolon. Hope this helps, brother Jim.

PS Be careful with commas; they are surprisingly easy to abuse.

Here's a short list (as you can see, a complete list would be impossible) of these adverbs, or conjunctions, or <insert POS here>s. It isn't really important how they're named; nomenclature varies from person to person like handwriting or vowel length. And there is no definitive list of Parts of Speech, anyway. What's important is how these thingies work.

also, anyway, as we were saying, consequently, finally, first, fortunately, furthermore, hence, hopefully, however, ideally, in spite of this, incidentally, indeed, instead, likewise, meanwhile, moreover, nevertheless, nonetheless, next, now, on the other hand, otherwise, regrettably, second, similarly, so, still, surprisingly, thankfully, then, therefore, third, thus, unfortunately, well, wherever we go,

Essentially, these are formulaic linking phrases (or clauses, or clauses reduced to phrases), which indicate how two clauses fit together in a context. Mostly those are, as you point out, independent clauses. However, independent clauses normally aren't marked especially. Ordinary sentences, for instance, are all independent clauses, but they have no "independent marker"; it's taken for granted, or -- as linguists call it -- unmarked in the syntax.

Normally it's the dependent clauses that need to be marked, lest they be confused with the unmarked independent ones. And also so that one has at least a prayer of discerning in which of the myriad possible ways the dependent clause "depends on" some other clause. There are a lot of kinds of dependent clauses, like complements and relative clauses, which have quite a lot of marking; and adverbial clauses, which normally use these thingies, too, though with somewhat different syntax.

Some examples of thingies that mark a clause as dependent, and also indicate the manner of its dependency, include:

after although how if once since lest because before than though till unless until when whenever where wherever whether while why

Notice no commas between. Generally, these are integrated into the clause syntax, rather than sitting out if front with a comma parenthesis like one finds with the linkers above. I've put all the single words in the list just above. There are a lot more thingies, but they're phrases. Some examples (separated by | because they usually don't use commas) include:

as if | as though | as long as | as much as | as soon as | as far as | inasmuch as | insofar as | even if | even though | no matter how | in that | in case | in order [that] | now that | so [that] | provided [that] | supposing [that] | given [that]

Notice how many as's and that's there are, and how many wh-words and prepositions are involved. These are idioms composed of spare parts and they're inserted whenever one needs a patch of a certain kind of meaning.

This doesn't really explain much about the various ways these thingies work, but at least there's a list or three of them. That's always a good start.

The word consequently is an example of what the Chicago Manual of Style terms a final conjunction:

Final or illative coordinating conjunctions denote inferences or consequences. The second element gives a reason for the first element’s statement, or it shows what has been or ought to be done in view of the first element’s content. The conjunctions include consequently, for, hence, so, thus, therefore, as a consequence, as a result, so that, and so then [he had betrayed the king; therefore he was banished] [it’s time to leave, so let’s go].

The function of a final conjunction is independent of its position in a sentence.

To understand what’s going on, let’s forget the punctuation and word order. Take the two sentences, The bus was late. Many people consequently missed their appointments. In the second sentence, consequently looks like an adverb of manner, an adjunct, modifying missed. But is it? Adjuncts take their meaning from within the clause in which they are found, as would be the case if we substituted subsequently for consequently: Many people subsequently missed their appointments. There, the adverb clearly modifies the verb and nothing else. It means they missed their appointments at some later time.

Consequently is different. It doesn’t tell us anything about the manner in which or the time at which people missed their appointments, but relates the second sentence to the first. Missing appointments is a consequence of the lateness of the bus and consequently is a type of adverb known as a conjunct. In the words of ‘The Cambridge Guide to English Usage’, 'conjuncts . . . play a cohesive role between separate sentences, or clauses. They . . . express logical relationships such as addition, contrast and causation.’ That seems to me an altogether more helpful description than 'marker'. In both of the OP’s examples consequently plays the same role. As a matter of style, I prefer the punctuation in (b).