There's a lot of FPSs that defined a certain period. (Geez, that sounded like the beginning of an elementary school essay.)
Let's try again with the power of MOVIE ANALOGIES:
DOOM is the uncensored exploitation movie of the seventies.
DOOM 3 is a PG-13 horror movie with really bad lighting.
Serious Sam, Timesplitters and Prey are the goofy 80s action-extravaganza that everybody likes.
Half-Life and Deus Ex are the smart, well written genre movie. Our District 9 or Inception.
Halo was the big damn Hollywood blockbuster. Avatar. Gladiator. Ben Hur. Caligula-... no, wait, not Caligula.
Quake 3 and Unreal Tournament are the sports movies or dance movies. Anything that focused on a particular activity at the expense of telling a story.
Modern Warfare and Bad Company 2 are the shaky-cam gritty summer flick.
Gears of War, Resistance and Killzone are the war movie where everyone speaks in grunts and homoerotic innuendo.
Bioshock is Ayn Rand's biopic.
EDIT: For the record, I've played and loved all of the above.