No. There are also amps that have a sliding class A bias system, where the bias is increased with the signal level. This allows them to idle without much heat. Krell used a system like this.
In tubes, you have class A1, class A2 (where grid current exists during part of the cycle of the waveform being amplified), class A3 (which was patented by Jack Elliano), class AB1 (no grid current) and class AB2 (grid current exists during part of the waveform cycle).
To be clear though:
There are amps that slide from class A to class AB (e.g. Pass XA25, Luxman).
There are amps that can be produced with different amounts of class A before transitioning to class AB (e.g. Coda).
-these are both simply class AB.