I'm not going to go as rules-heavy as Snarf. You can define a trilogy for yourself, but for ME, I think to qualify the subsequent installments need to have been intended as sequels to the first and second. So this rules out the "Dollars" trilogy of Eastwood movies, which Leone intended to be stand-alones. (Convenient for me, as The Good, the Bad, and the Ugly is probably my favorite film).
I feel like this is a distinction without a real difference when you're including Indiana Jones and The Dead, Mad Max and so on, all of which were essentially made as stand-alones with no specific intentions of further sequels or even exploring that subject further, and which all work as stand-alones. Literally the only difference what the author said in an interview. Functionally I would go as far as to say the Dollars trilogy is more of a consistent trilogy, on every possible level, than say, The Dead or Mad Max, and I say that really liking all of those movies.
I'm not going to argue that specific point further but it seems like if all you're going on is "the author said" like that, you're creating a rather unnecessary and hugely arguable situation. Like planting landmines in your own garden really.
I think my personal issue with lists of trilogies is that 90% of "trilogies":
1) Have at least one film that kinda sucks or even absolutely sucks in them or at least wildly worse than the other two. This includes most of the ones people are posting, honestly.
Like Austin Powers come on, the third movie is awful even by Austin Powers standards. Hilariously it's basically the same problem as The Godfather, Part III (which if two movies can be good enough to make up for one bad one, should put The Godfather on way more lists than it's on), which is that the director no longer has people saying "No, that's a bad idea" to him, and has become increasingly self-indulgent and pampered. Hell even the Cornetto trilogy has that problem, in that The World's End is drastically worse than Hot Fuzz or Shaun of the Dead, and absolutely only exists because of own-koolaid-drinking. It's like 5% as funny (seriously) as Hot Fuzz, and it's not even as emotionally resonant! And I watched it exactly the age it's supposed to be about! I like that it's anti-nostalgia, we need more of that, but you can do better than that, man.
2) The series actually has more than three movies, so you're kind of just picking the first three arbitrarily (which may exclude the best entry, as it does with Mad Max, or the worst entry, as it does with Indiana Jones).
So my personal rule is I won't include those as "best trilogies", which means I gotta go only with movies that actually have three, and none of them even kinda suck.
So:
1) Lord of the Rings - obviously. I didn't even like LotR before I watched the movies, though I then went back and re-read the books and so on.
2) A Better Tomorrow - Yeah all three rock. Good luck finding them on streaming in most places though!
3) Three Colours - There's one I don't like, but it's absolutely a good movie. Red is by far the best though.
As I am bound by my own, self-imposed chains, as well as your chains re: "well the author has to say it's a trilogy", I'm not sure I can think of much else, where there's neither one that that suck, nor are we cutting off further (usually bad) sequels. Maybe Bourne? Maybe the recent Planet of the Apes movies (certainly not the originals lol)?
If I ignore the rules I self-imposed, and yours, then Mad Max 2/3/4 (not that one is bad, but we presumably have to pick three contiguous movies, and also they're all more similar to each other than to one), Die Hard 1/2/3 (let's pretend that where it stops), The Hunger Games (super-cheating lol), and obviously Dollars would be #1 by a mile.
