idTech 2 is the Quake engine but it's also Quake 2's engine (but, as Snehk said, it was modified for it).
id Tech 1 is Quake, id Tech 2 is Quake 2, id Tech 3 is Quake 3, id Tech 4 is DOOM 3, and id Tech 5 is RAGE. DOOM's engine (as well as Wolfenstein's engine) used for the first two games was never really given a name since it was only licensed out to a select few companies. Some call them id Tech 1, some call them id Tech 0, and others call them id Tech, but they're really just called by the name of the game that used them. The "id Tech" names only came about simply for the convenience of licensing and making the engines sound less specific to one game.