The term "Trilogy" is connected to the way plays in Ancient Greece usually came in 3s (or 4s with the 4th being a Satyr play). But I feel to modern audiences the definition of a Trilogy comes from the original Star Wars trilogy more then anything else.
Certainly no major Film trilogies predate Star Wars, The Godfather didn't become a Trilogy till after, and Coopola didn't even want to call that one Part III, it was never made with a Trilogy format in mind.
In Prose fiction there were Trilogies, but many only line up to some aspects of what Star Wars makes us think of as a Trilogy's rules. LOTR was really one big book published in three Volumes, C.S. Lewis did not originally plan out of the Silent Planet to have sequels at all. But again perhaps the first modern Prose Trilogy was Alexandre Dumas' D'Artagnan romances.
Lucas was trying to follow the theories of Joseph Campbel, so certainly his structure had a basis in what came before. But it's also clear he wound up making conventions as well as following them.
Mostly the Trilogy convention I'm talking about is the assumption that the Second/Middle Film must be the Darkest. And even when someone isn't going for a Trilogy they still feel the need to make the second film waaaay Darker then the first. That's why The Dark Knight and Batman V Superman are both very dark in their own different ways. And every time someone wants to hype up their second installment as being oh so dark they cite Empire as the precedent, and also Wrath of Kahn, a movie that was arguably Empire's first imitator.
First I find it an interesting overlooked lesson for the narrow mindedness of modern writers to point out that The Empire Strikes Back happens to kill less people then either it's predecessor or follow up, yet is undeniably darker. ANH kills Owen and Beru, Ben Kenobi, an entire planet right in-front of a woman who grew up there. And then all but one villain between the Deathstar blowing up and what happens to Greedo. And ROTJ kills Yoda, and all it's villains, one right after being redeemed. But Empire kills only random war casualties, and Imperial officers Vader force chokes after appearing in only 1 or 2 scenes, the one officer we spend enough time with to maybe care about gets let off the hook at the end. And no notable good guys are killed, the most is one major character being captured.
So I'm tired of modern TV writers going all "if we don't kills someone every other episode, or at least once a season, how will the show have stakes?", TESB had high stakes without killing anyone.
But back on topic.
I find it notable in light of that being so central to Empire's legacy, that the most notable Trilogy to defy this, "the Second must be the Dark one" structure, is the Star Wars Prequel Trilogy. Lucas defied an expectation he himself set, and I think that is at the heart of why the Prequels are so unfairly hated.
Now the "Evil must prevail in the end because it's a Prequel" premise could be said to be why it was unavoidable that this time the third film would be the Darkest. But it's interesting that AOTC doesn't try to be noticeably Dark at all. I mean it is probably darker then TPM, but in a lot of ways it's more reflective of what a Sequel was expected to be before Empire/Jedi defined the modern Trilogy.
In a lot of ways The Prequel trilogy follows the pattern of LOTR more then it does the previous Star Wars trilogy. Return of The King in both books and films has the Darkest chapters of the saga, though it having the ending may balance that out, but the ending is bittersweet. ROTJ had it's dark moments, but mostly those are only superficially dark, wanting us to think Luke could fall so that it's a victory when he doesn't.
TPM like FOTR is what one would expect from their relevant Fantasy sub-genre, (though TPM is more capable of working as a stand alone), and both have allegedly "boring" council scenes. Then the Middle installment is about a War breaking out, while in the third the War ends. Also ROTK and ROTS both reach their climaxes in a Volcanic setting, right from the teaser it was clear Mustafar would have a Mordor vibe. And Frodo and Anakin are both consumed by The Dark Side in the Crack of Doom.
I also do not think it a coincidence that Saruman and Dooku were played by the same actor, Dooku I think was intentionally a hybrid of Dracula and Saruman. And both are killed early in the third film.
Another Trilogy that perhaps helps explain why The Prequels are so different from the OT as a Trilogy is the D'Artagnan trilogy. That trilogy also ends in a Dark Place, but with a promise of A New Hope, that was for Dumas the coming French Revolutions. Aramis also goes from being a Hero to a Villain by the end.
And now how should these observations help us evaluate the New Trilogy that just started? We only have one film right now, but expectations are that this Sequel will again try to be Empire/Wrath of Kahn/The Dark Knight. Whether or not that is how it must be is perhaps a difference between Prequel fans and Prequels haters.
The reaction to BvS has shown that audiences are getting tired of every Saga being a downward spiral (and Suicide Squad's unanimously good word of mouth in-spite of bad critical reviews shows you can't blame BvS's failure on anything else about the DCEU besides being dark). And the legacy of Temple of Doom shows that a Lucas Film property isn't always praised when it goes dark. So maybe it's time to do something different from either Lucas Trilogy.
The Force Awakens actually is dark enough that we could see a Trilogy where the first installment is the darkest for a change. TFA has been called the darkest of all seven movies currently, but you'll have trouble convincing me it's darker then Sith. Still it's certainly a superficially Dark film, one that entered production when The Dark Knight Rises was big and being Dark in general was still the in thing with only Marvel defecting from that. All three first installments kill the mentor figure, but Han's death is able to carry an emotional wait that Qui-Gon and Ben Kenobi's could not.
I don't think a franchise with "Wars" in the title should ever stop having some dark themes, but if Episode VIII turns out to be done with the objective of being the Darkest thing ever, at Christmas time, I really don't think it'll go over well.