When should end punctuation go inside quotes?

I have been/am being taught that end punctuation should always go inside quotes. For example, you are supposed to write:

Marvin thought it was "awful."

The problem is I do not see how does this make sense. Intuitively, I always wrote:

Marvin thought it was "awful".

as that makes more logical sense — you want a quote to be an exact replication of what somebody else said, so why should you add punctuation inside?

I always thought it made more sense to not touch the quote and add anything after or before if it must be added.

So, why should I put end punctuation inside quotes?

Firstly, this is only American convention — in Britain for instance you wouldn't use it (except for a few publishing houses). Secondly, this is not logical but typographical: a convention arising out of early American printers' opinion that typesetting the punctuation inside quotes looked better. This convention is slowly eroding in some areas and being replaced by the "logical" one… but it is still the predominant American convention. English is made up of a great many mere conventions and you can't really demand that it be logical.

It makes a bit more sense if you consider quotes that contain several sentences.

He said, "This is a sentence. This is another. All sentences have their punctuation inside the quotation marks."

It does seem to make sense when using quotation marks to delimit a single word to place the punctuation outside the quotation marks.

My password is "foo.bar.".

For reasons that are unclear to me, the title of this question was edited from "Why should end punctuation go inside quotes?" to "When should end punctuation go inside quotes?" These are different questions, and I don't see any clear answers on this page to the second (current) one.

The rules about when to put punctuation inside or outside of quotes are fairly complex, particularly if you are using "American" or non-"logical" style. Not all punctuation marks are placed before quotations in American usage: for example, you wouldn't write

❌ Did Marvin think it was "awful?"

You should refer to a style guide for guidance on when to place a period or comma before quotation marks in American style. In a separate answer post on this site, Barrie England recommended Larry Trask's treatment of this subject, linking to a document that unfortunately is no longer available at that address. Wikipedia has some coverage of this topic in the article "Quotation marks in English".