I never use the pressure relief valve to "fill" my water heater tank and my water pump does not run when water is not called for.
Use white Teflon tape on the threads of the anode.
What valve are you opening to fill the tank? If it is the water heater bypass (usually added later by an owner) installed right at the water tank, there are two valves, one at the top of the tank and one at the bottom with a hose between them. This is for winterizing your vehicle with the pink antifreeze. These valves are used to divert the water around the water heater so you don't have to add those 6 extra gallons of antifreeze. Be sure you turn both valves , never just one whenever a change is made. If you do winterize, remove the anode rod to drain the tank. I usually leave my rod inside the outside door to the tank.
If you need to figure out which way the valves should be turned, turn off the pump and open a cold (and hot) water tap to release pressure, close taps(s). Remove anode rod and turn on pump. Check for water coming out the anode rod hole. Change the valve positions at the water heater until the water runs out the hole. Usually the lower valve is the cold water inlet to the water heater. After you have water freely flowing out the anode rod hole, shut off the pump. Put the anode back in using white Teflon tape. Turn the pump back on and open the hot water faucet in the sink. Air should rush out as you are displacing it in the tank with water. If air does not rush out of the faucet, turn the other valve at the hot water tank till it does. Your valves are now set to normal use mode. Turn both valves the other way to set it in the bypass mode.
Open a hot water tap until water runs free, no surges. Same for the cold side and do all of them including the outside shower. Now look for leaks if the pumps still cycles.