Err... almost. An engine is something that has significant engineering effort put into it, you can see that the words are cognate. A motor is (usually an engine) that converts potential (stored) energy into motion, by way of some force. I do believe that motor and motion are also cognate.

But outside the etymologies, there is no standard, agreed upon definition for either the term engine nor motor. I personally like these etymology-based definitions (otherwise how do you excuse the term "siege engine") but it's not a hill I'd die on.