Plenty already mentioned so I will skip them.
Far Cry though that might be more a case of someone buying a name. That said if we consider the first as something else then more has built off 3 than they have off 2.
Bomberman maybe (you had the 80s one, the 1990 one, II kind of was its own thing and then we hit the 16 bit era stuff which inspired most takes to this day. If you had one on the NES and then on your 16 bit console as your second bomberman (or maybe you went PC and Atomic Bomberman) and have played anything save perhaps the N64 efforts and the awful act zero then tell me it is not that.
Jedi Knight/Dark Forces. While 1 was a considerable hit and referenced to this day it was the second that gave us most of the rest of the series.
Depending upon how you want to count Zelda (basically you are going to want to skip the sidescroller, which many in the west but in Japan it is apparently legendary). 1's puzzles and setup was inspirational to the later games but at least everything since then in the 2d side of things has been based on the 16 bit style, aesthetic and flow.
Splinter Cell perhaps though that is less clear. Might just be unrefined more than substantial differences. That said the more free form thing rather than almost a real time point and click did start with 2.
3d GTAs maybe. 3 was good but Vice City set the template for everything since. That and I will have an easier time drawing a line from the general world feel of II to the post 3 3d series (however I mostly blank the Londons from memory so who knows, or maybe we can write them off as expansions/dlc).
Final Fantasy, at least until 12. Go play 1 and it feels kind of odd where 2 is vastly unrefined but unmistakeably Final Fantasys after that until 9 and pretty clearly a heavy influence for 10 and 12 even if that is where things started to diverge a bit.
Carmageddon. Not quite as clear cut as some but 2 went on to do anything else I have seen from the series. Not so sure it is a good thing either, though the more recent effort I reviewed a while back did seem to lean back towards 1 which I appreciated.
Old school Tomb Raider. 1 holds up as a game more than some of the others but II was what everything that followed was about.
Resident Evil 2. At least until 7. Everything else seemed to have 2 as a jumping off point for gameplay more than 1. Probably the only one on the list as well that has constantly been returned to (how many remakes of 1 have we had now?).
Devil May Cry perhaps. Though I am less familiar with later games in the series. I do however remember playing 1 a lot and then 2 when it hit. Everything I saw and played of later offerings was this.
Advance Wars. Now technically there was famicom wars and super famicom wars and gameboy wars but the move away from more rock-paper-scissors type play to everything bundled and have a good time started properly with the crazy tanks and leaning into COs in that.
Tony Hawk 2. Here it went from kind of silly but at some level believable to crazy scores. On the subject of skateboarding games I would imagine Skate 2 will probably be the series defining game for many but we only had one sequel and some DLC so eh.
Borderlands. 2 pretty much set the tone for everything since, though that might not be a good thing. 2 and beyond might have better mechanics (though I could see some argument about regression in some ways) but I would take the tone and humour of 1 over anything else I have played since.
Heroes of Might and Magic. Some might say King's bounty and they could make that argument but if we are going strict naming then 2 (or at least 2 with expansions) led to 3 more than 1 ever did which is what everything else has tried to get back to.
Need for Speed. Second series. Started with Hot Pursuit but then we got underground. While most would probably look at underground 2 as the main example it was those that everything since (barring things like shift) have been trying to get back to and usually failing at some level.
Elder Scrolls. Daggerfall led to Morrowind and then clear line to skyrim from there, though I would note the massive simplifications from Daggerfall to Morrowind and then onwards. Arena? What is that.
Age of Empires 2. I know the series went rather off the rails (even more so if you count the DS efforts, though they do have something to them) after 2 but they all clearly wanted to be 2 rather than 1, though 1 is still very good.
Just Cause 2. 1 is clearly recognisable in the rest but 2 is what set the scene.
Edit.
Diablo 2 maybe, I guess that game kinda defined the entire genre
I don't know if I would go that.
More classes, few more moves and more overworld stuff than straight dungeons but 1 and 2 play so similarly that I can't discount 1 and I dare say most people would not either.
Or if you prefer make the graphics the same, condense the game down a bit and could you call 1 a training mission/level/gameplay arc and not have too many people notice much wrong compared to some of the other stuff mentioned if you did that with its sequels?
