Well. when the water heater kicks in the water in the tank expands, this will increase the pressure, if there is no check valve on the water heater inlet (no reason there should be one there) the pressure on the city water inlet will increase.. As for the water pump being bad...... there are a couple of types of water pumps and the more common one can NOT fail high pressure, the more expensive one can.
If the pressure got high enough, theory has it it could blow the check vavle in the city water inlet,,,,, but the TPR on the water heater should go way sooner than the city water inlet.
Soem folks say "Do not ever press on the white pin in the center of the city water inlet (you have to remove the screen washer to see it"
Well.. if there is any kind of debris in the valve, pressing this pain may allow it to flush out. and allow the valve to re-seal.