They won hundreds of dollars; five hundreds to be precise!

While it may be O.K., "five hundreds" is not usual.

The place of a digit in the number is called "units", "tens", "hundreds", and so on; but when talking about quantities, the only one to become plural is the unit: 1 unit, two or more units.
The author you quote mentions "grammar books" recommending not to use the S, like this one at Woodwardenglish:

When we have large numbers or a specific number, we do NOT put an S at the end of hundred, thousand, and million.

For example, in this report at unesdoc.unesco.org, the number is written without the S at the end of "thousand" and "hundred", just like the one "million", although they are plural (Deakin mentions this custom rather briefly in the paragraph that you quoted):

[...] before the end of 2003, the sum of one million, two hundred and eighty-two thousand, five hundred and fifty-five United States dollars (US $1,282,555) corresponding to the [...]

Constructions like "two hundred tens", while mathematically decipherable, are definitely uncomfortable, to say the least, and smack of gratuitous polynomial expansion.

From the use in one of the examples at the entry for "hundred" (OALD) you can see that an s is probably not used. From this ngram and this one you have the confirmation that an s is never found after a number; it must be exceptional to find one.

First, I will answer your original question. Below the answer, you will find all of the information I have found about other usages for "hundreds" (there are probably some that I missed).

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They won hundreds of dollars; five hundreds to be precise!

As the sources below state, the author—technically—should use "hundred" in this case. However, the author used "hundreds" in the previous sentence, so they had the stylistic choice of whether they wanted to add the "-s."

In this case, the author was likely showing that the money was won in discrete units of $100. On the other hand, the author could have simply been matching the "hundreds" from the clause before the semicolon; since they are in the same sentence, it would make sense to match the plurality of the same noun. It is simply a stylistic choice by the author.

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I apologize for the long(ish) answer, but you have included (since your edit) many examples, so I will give you all of the information I have found.

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From A Comprehensive Grammar of the English Language (p. 308):

The nouns dozen, hundred, thousand, and million have zero plurals when they are premodified by another quantitative word:

three dozen glasses

two hundred people

The plural form is normally used with all four nouns when an of- phrase follows, with or without a preceding indefinite quantitative word:

(many) dozens of glasses

(many) hundreds of people

But the zero form is common enough:

a few million of us, several hundred/thousand of them

Note such combinations as:

tens of thousands of people

hundreds of millions of stars

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From A Comprehensive Grammar of the English Language (p. 396):

We always read year date as hundreds:

In 1985 'nineteen eighty-five' [or] 'nineteen hundred and eighty five' <formal>

In the 1600s 'sixteen hundreds'

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From The Cambridge Grammar of the English Language (p. 351):

i a. dozens of spiders b. hundreds of voters [head noun + complement]

ii a. a dozen spiders b. three hundred voters [determiner + head noun]

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From Michael Swan's Practical English Usage (p.322):

After a number, the words dozen, hundred, thousand, million, and billion have no final -s, and of is not used. This also happens after several and a few.

Compare:

five hundred pounds

hundreds of pounds

several thousand times

It cost thousands.

a few million years

millions of years

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From Cambridge Dictionary:

There were hundreds of people at the pool today.

There were a hundred shirts waiting to be ironed.

He expects the total amount to be in the low hundreds.

The house was built in the sixteen hundreds.

Breakfast is at seven hundred hours.