I do not use the city water connection. I understand as the water comes in from it there is a check valve. If the pump is turned on and the water pressure goes below 40lbs it will turn on.
If you were filling the tank while on city water. Some valve in the pump was not doing what is should do. That is, prevent water going through the pump into the water tank. Who knows, it could have had something holding a valve open.
If the pump works normal without city water I would not worry about it.
I know nothing about drain taps mentioned above.