I guess it can go either way. After all, we put "motor oil" into our gas engines. And what about gas-engine powered motorboats?
But I always thought there was a technical difference between and "engine" and a "motor," although I don't know exactly what that technical difference is! I think it might be that an "engine" needs a separate fuel source (gas, diesel, water for steam, etc.) while a "motor" doesn't need to be fueled, because the "fuel" is electricity.
Also, an "engine" converts thermal energy to mechanical energy, a "motor" does not.