Arena shooter is descriptive of certain mechanics. Armor/weapons with tinted spawns on the map, usually some kind of bunny hopping, but that's not mandatory, and often portals/verticality.

I'm talking about quake, unreal, warsow, xonotic, etc...

I think what you guys are on about are games like doom, Wolfenstein, Duke nukem and so on. Not so much quake but single player.