Particularly if you have a three-valve water heater bypass, having the valves in an inconsistent setting can lead to some strange behaviors. I once managed to open the cold water inlet valve but not the hot water outlet valve and not shut the bypass valve, which made the water heater act exactly like a six-gallon pressure tank. The pump would run for a long time, and then when shut off there would be water flowing for a good length of time afterwards.
It may be that the pump isn't self-priming for some reason. Hooking up the city water may help with that. Filling the water tank all the way up, if it's not already, may also help. Starting it from the winterize inlet also might help if that's the problem.