Good question, Ben. I always called that thing the cylinder, but I'm not sure that's correct for air pistols.
I'm also not sure that, given the pellet shape, and the low power level of air pistols, there'd be sufficient recoil or gas pressure to work a cartridge-gun type of mechanism, that would reload a chamber and cock a hammer.
At least, there's be no need to eject an empty cartridge casing.
Cheers!
Jim