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.
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.