Having the city water connected and the pump on is ok, if you have water in the freshwater tank.
This happened to me. We were connected to the city water and no water in the FW tank. The wife turned the pump on while I was out walking the dogs and it ran for a good while as it could not pressurize using water from the FW tank. I came back and heard the pump on and immediately turned it off. A few day's later I was able to add water in the FW tank and turned the pump on. It did take it longer to charge the system but it does work. I'm surprised it didn't burn up.
Easy enough to tell if it burned up. Just put some water in your FW tank and with no city connection, turn on the pump. Does it pressurize the system and does the pump turn off automatically? If you turn a faucet on does it provide water ok?