I certainly default to assuming it does and in many contexts it is an explicit rule.
I don't understand how you've refuted that while also saying they sometimes don't. Are we arguing about contexts here? My claim is almost trivial - nobody reasonably familiar with English thinks quotes imply a verbatim quote. That's just not what quotes are for.
Getting back to the actual point, in formal writing, quotation marks are definitely considered to delimit actual quotes. That's where their name comes from and that's their purpose. If you want to paraphrase or otherwise interpret what was said you just work it in without quotes.
Personally, I relax my expectations in informal contexts if I don't know the person or their writing habits, but I'm just being pragmatic. In other words, the rule doesn't change, it's just not always followed.
That your position is that I’m in the minority on this is doubly surprising to me given that’s what all the style guides and my high school English teachers taught me.
"In English writing, quotation marks are placed in pairs around a word or phrase to indicate:
Quotation or direct speech: Carol said "Go ahead" when I asked her if the launcher was ready. Mention in another work of a title of a short or subsidiary work, like a chapter or episode: "Encounter at Farpoint" was the pilot episode of Star Trek: The Next Generation. Scare quotes used to mean "so-called" or to express irony: The "fresh" apples were full of worms."
Even 'direct speech' is at odds with 'verbatim quote' and that's the first thing there. Direct speech can be completely made up.
"AP, MLA & CMS" are an absurd counterpoint that falls well within 'that's not how anyone writes'. They are, if anything, lengthy exceptions to how anyone writes.
It's a deeply silly argument and my point is 'an internet messageboard should not be regulating punctuation'. It should, as this one usually does, try to regulate behaviour.
(For what it's worth: this little subthread is about 10x more interesting than the story and the rest of the thread it's attached to).
Don't be an ass.
Don't call other people asses.
Don't complain about votes.
And then:
Some weird thing about quotes we can't even sort out as well-intentioned nerds who love to talk about rules.
I don't think that's a good rule. I think what it's trying to address is probably a good rule. But it's addressing it in the dumbest possible way.
Now, I think that it’s a fair argument that a web forum needn’t have the same formality as other written word, but your assertion that “it’s not how anyone writes” is clearly untrue.
And just as a single data point, I expect when someone uses quotes even on the web that they are asserting a verbatim quote.