You are describing either low fluid or a bad pump. Now, again, all I have to go on is your text. If you made a video of the entire process, maybe I'd say something different, but based on what I read, I'd say low fluid or a bad pump.

ANd running the system on low fluid for a time CAN, I suppose, damage the pump - but it would take a LOT of top operations to do so, if it can be done. I say this cause a lot of us have had fluid leaks and low fluid, as the cars are getting old and cylinders are starting to leak.