No such thing as a 'hot water heater,' Hot water doesn't need heating.
The tank should be full before it's used. It has space for air in the top of the water heater tank. Air can be compressed and water cannot. The air is needed to absorb the expansion of the water as it's heated. With the water pump or city water off, opening the pressure relief valve will allow excess water to come out and air in to make sure the tank has the properly sized air in the top of the tank.
If you're hearing the water flow, it could be a calcium build up in the check valves that prevent backflow in the water system. If you only hear it when the electric heating element is on, again it could be deposits on the heating element. Might be time to clean the water heater with vinegar. (YouTube has video how-tos)