Hey, you wanna throw in SS with Doom go right ahead, I refuse.
You could just say "Doom clone" if you mean Doom clones. But the point is that nobody's going to be able to come up with a definition for "FPS" that covers games like Doom, Half-Life, Metro, Far Cry, Rainbow Six, etc but manages to arbitrarily exclude shit like SS.
To desperately try and bring it back to the thread topic, maybe this is part of what causes the difference between 90s FPS games and throwback games - devs at the time were just seeing what you could do with the formula and trying all kinds of different things and genre fusions, since the term "FPS" didn't exist and nobody gave a shit about matching some undefined set of criteria, and thus there was a lot of variety and experimentation, leading to games that are difficult to group together under any specific definition (Doom, Hexen, System Shock, Killing Time, Realms of the Haunting, Strife, PowerSlave, Unreal, Half-Life, etc). Meanwhile throwback devs are setting out to very deliberately create something that they think of as an "FPS game", which to many of them seems to mean a specific style based entirely around particular aspects of Doom and Quake.
And to many of them that seems to translate to "everything looks like shit, environments are abstract with big open areas, waves of enemies are attacking the player, no plot or cohesive visual design, lots of jumping around, shitty heavy metal blaring out".