In my mind - the pump must chamber the ball, and recock the marker without assistance.
If you only chamber the ball by pumping it - it is not a pump.
If you only re-cock the marker by pumping it - it is not a pump.
I think of this like a pump shotgun. It both recocks the shotgun, and chambers the round.
If you can fire the marker repeatedly without pumping it - it is not a pump.
To me, many markers depend on if you deem loading a spring to be essential to it being a pump.
P-mag's require you to chamber a ball (or allow a ball to fall into a chamber) and reset the marker to be fired. But you are not loading a spring.
MQ'd pumps require you to chamber a ball - but do not reset the marker to fire. I disagree with Diomedes here - you can fire again without pumping - you just cannot fire a ball. It, to my mind, is not a pump.
PA'd pumps are just auto cockers that toggle the 4 way with a pump handle. They are the least pump-like of the breed.
With all of that said - I really don't care what I play with or against - pumps are hard enough to play with - so they all 'count' in my book - but some are not pure pumps.