You got it - the action is the determining factor.
When it's stamped 22 S,L,or LR it means the gun will function/feed with all three types. If it's stamped 22LR only it means it will only feed properly or operate the action with 22 LR ammo.
For example the Ruger 10/22 is designed to operate with 22LR ammo (to fully cycle the action), but you could fire 22 shorts or longs in it by manually loading one round at a time - turning it into a really awkward singleshot!