Electric motors have moving parts and at least where I am nobody would call one an electric engine.

Automobiles generally have engines, but motor isn't uncommon for (internal combustion) engines there either. Motor car is something I remember my grandfather saying although I guess that's not really used anymore, but motorbike certainly is. So is motorsports. "Blown motor" is a pretty common phrase for an IC engine that has stopped working due to damage.

Engine has some exclusive cases where motor does not substitute. Jet engine and steam engine come to mind.