"Based on a true story" is one of those film-marketing buzzphrases that means almost nothing. If the initial inspiration for the story is a true story, even if it bears little resemblance to the final product, it is "based on a true story". See also: Subway's "made with 100% chicken"

"Based on a true story" is not the same thing as a documentary; it's assumed that aparts of the story are not real; how much is made up is up to the writer, not the reader.

Not sure what you point is. Fargo says "This is a true story".

It adds that apart from the name changes "the rest has been told exactly as it occurred."

A blatant lie, not at all in the manner of "based on a true story".

Fargo's credits state:

"The persons and events portrayed in this production are fictitious."

Still not sure what your point is. You seem stuck.

Nobody reads the tiny print in end credits. Viewers were told in opening credits that every detail except the names was true.

Someone went looking for the buried money and got themselves killed as a direct result of the statement in the opening credits.

Every single modern movie has that lawyer-require statement just in case they get sued.

No, real (by intent and marketing, not necessarily truthfulness) documentaries (which are modern movies) do not.

Movies that are actually intended as works of fiction (even if inspired by real events, and even if—well, given the purpose, especially if—including some characters retaining the names of those in the real events doing some of the things they actually did) have it.