Upvote for the main thought, but

1. A gasoline engine is a type of motor. You are making a distinction without a difference. A motor is just something that produces mechanical motion. Motor. Motion. Motivate. All come from the same root as “move”
2. An “engine” can be the whole apparatus that does the work, not merely the thing that provides the power, and could be powered by a water, beast, or even manually such as the cotton [en]gin[e]. A lot of that usage comes from pre WW2, but it survives in things like “train engine” (the vehicle that pushes or pulls a train) and “game engine” (the program architecture that manages the processing power provided by the computer to convert user interactions, game assets, etc. into something that we recognize as a game).
3. It’s not a four-cycle engine. It is a four-stroke engine: each piston makes four strokes each complete engine cycle.