This is a great example of why sometimes there is no "right" article to use, because either one can work.

Based on a true story = part of this movie really happened.

Based on the true story = part of this movie really happened.

Let's say I produce a movie called Maulik's Adventures on the Stack Exchange. Is that based on a true story, or based on the true story?

Well, there are many true stories, but only one true story that my film is based on, so the works just fine:

the (article) used with a qualifying word or phrase to indicate a particular person, object, etc, as distinct from others ⇒ ask the man standing outside, give me the blue one

That said, we are talking about a particular story – one story, the story of Maulik on the Stack Exchange – so a works, too:

a (article) An adjective, commonly called the indefinite article, and signifying one or any, but less emphatically

So:

Come see Maulik's Adventures on the Stack Exchange, based on a true story

means, "this movie is based on one true story" (but expressed a bit less emphatically).

and:

Come see Maulik's Adventures on the Stack Exchange, based on the true story

means, "this movie is based on the true story of Maulik" (but not the true story of StoneyB, or J.R., or ColleenV).

Your bewilderment stems from an erroneous belief that there exists some litmus test you can use to figure out when you should use one article or the other*. Not so! Macmillan lists 15 possible meanings of the word a, and 14 meanings for the word the. There are some contexts (such as, "Based on _____ true story") where you'll find a meaning for either article that would indicate a valid usage, so either article can be used.

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^{*Usually this "rule" goes something like this: "If you have referenced it previously, use the"}

My guess would be that "based on the true story" means "based on the story we all know". It could also mean "based on the only true story: we've been meticulous in our effort to be as close to the truth as possible". It could be a way to whet the viewers' interest, a kind of a brag, an overstatement.

The second sentence, "based on a true story", could mean "based on a story you (the viewer) don't know yet." Maybe because it was a state-guarded secret. Maybe because it was not as widely known as other stories.

Based on the true story of that famous murder that happened back in 1996.

and

You'll see a movie about Alan Turing. It's based on a true story.

Note that I postmodified "the true story" but left "a true story" without further modification. Had I continued: ".. on a true story of the cracking of the Enigma", one implication would've been that there are possibly other true stories of the same.

When I do a search for "based on the true story of", I get some 240 results at Gooble Books versus about 70 for "base on a true story". The definite article seems to go better with more details (the story of what/who?) than the indefinite.

Here's the Ngram for "on the true story of / on a true story of".

One thing that bothers me is that the COCA corpus does have a sprinkling of examples (9) of "based on a true story of", with this, for instance, being quite speficic:

Photograph Hounsou (2nd, l), along with the talented cast of African actors, delivers a compelling performance in Amistad, based on a true story of an 1839 slave revolt aboard a Spanish ship.

Maybe this means that several stories can throw light on the events differently, from different angles, but nonetheless be "true".