Gen X here. Arriving late to the thread but I think I have something to say about this. I’ll try to keep the list short, as it could go really long if I added everything that deserves to be mentioned. Games that were genre-defining for me:

• Gradius/Nemesis (this saga defines what a good shoot 'em up should look like, noteworthy title being Nemesis 2 for the MSX)
• Double Dragon (original arcade version, as it was never correctly ported to any other platform due to home consoles being too weak at the time, this is the beat 'em up game you must play)
• Shadow Dancer (for the Genesis/Mega Drive, the best title in the Shinobi series, has great graphics, music and gameplay)
• Bionic Commando (for the GameBoy, there’s just something special about this game that was never truly replicated)
• Super Mario World (the Super NES title is 16-bit platformer perfection, the interaction between sprites is just what you expect it to be)
• Ninja Warriors Again (for the SNES, is just a fun, very well-made beat 'em up)
• Street Fighter Alpha 2 Gold (a.k.a. Street Fighter Zero 2 Alpha, this is 2D fighting perfection, simply put)
• Castlevania: Circle of the Moon (the series’ debut on the Game Boy Advance was also one of the best games ever made for the console, arguably the very best 2D title in the series)
• Doom (the game that made Boomers get interested in video games for better or worse, requires no further introduction)
• Ōkami (this one has been re-released for every platform since its original PS2 version and if you play it you’ll understand why)
• Castlevania: Curse of Darkness (hands down the best 3D Castlevania title ever created, criminally not ported to newer consoles)
• Resident Evil Zero (probably the best fixed-camera title in the series, also sets a good basis for playing the next ones afterwards)
• Resident Evil 4 (the original, not the remake, arguably one of the best video games ever made, also first “action” title in the series)

Anything that comes after that is probably not “old” enough for me.