Errr no they didn't All those games you listed were mods created by end users for the Half Life engine.
Valve created Half Life, which was at the time the greatest FPS going... by the tiem Half LIfe 2 came out they had employed many of the people making those mods.
FPS online gaming really started with Quake (especialy when the 3dfx addon card came out) and grew from there... back in the day Quake 3 was the big seller, where people like fatality made his name...