The description of TF2 on the Orange Box blurb has a typo. The original Team Fortress did not spawn the team based multiplayer action genre, it spawned the class based multiplayer action genre.
Prior to TF, you started with no weapons, and had to run around and get a weapon first. TF was the first multiplayer game to give you all your weapons when you spawned based on your chosen class, and to have different classes be so different as to be playing nearly different games.
Team Fortress's innovation was class based gameplay, not team based gameplay.