The chamber for a 22LR is longer than a 22 Short, but not fatter. The chamber for a 22Mag is fatter than a 22LR. The bullet itself is actually slightly larger too.
Although quite similar, they are not in the same family of rounds.
Well, krochus said it better than me.
I'll add that 22LRs rattle around in the 22Mag cylinder of my Ruger Single Six Convertible, and 22Mags won't even fit in past the bullet in the 22LR cylinder, even though both rounds shoot down the same barrel when using the appropriate cylinder for the round. It's also my understanding that these guns have a bore that is oversize for 22LR.