"I started playing my first FPS game, Quake 3: Arena, in June 2000."
Admittedly, the wording is a little confusing and misleading. This document attempts to explain the wording.
The very first FPS game that I actually played at all was Wolfenstein 3D. This was, however, on someone else's computer.
Through the years, I also played smatherings of Doom 2, Dark Forces 1, and Quake 2... on other people's computers. In the meantime, the majority of my gaming was spent on old DOS games (Scorched Earth!), old Windows games, my dying NES (Super Mario Bros. 1!), NetHack, and the local video game arcade (King of Fighters 96!).
Not until June 2000, with an upcoming LAN party, did I go for my very own license[d copy] of Quake 3, thereby becoming the first FPS game that I could play at my own leisure on my own computer. Thus, "my first FPS game".