Why is it that when the engine in our car won't start we get out and check the motor?

I know my car has a starter motor and a windshield wiper motor, but it also has a "CHECK ENGINE" light. And in my Webster's, no snickering from the crowd among the various definitions of each, I find:

engine n. 1. A machine that uses energy, esp. energy from a fuel, to perform work.

motor n. 1. A machine that converts electric energy into mechanical power. 2. An internal-combustion engine.

Are the first definitions the same or different here?