Rocket motors don't convert electrical energy into anything. Motorways are ways (roads) for automobiles, most of which which use petroleum products for their power. Motorcycles use gasoline to produce their mechanical motion.
I view "motor" as a subtype of "engine". An engine drives a process or system, either literally (producing power from some fuel or other energy source) or figuratively (as in Google's search engine). A motor is a type of engine that changes the momentum of some object.
If you use the expanded definition of engine (e.g., "search engine", where "engine" is mostly used as a synonym of "machine"), then neither is a subtype of the other - they just overlap occasionally.
However, all engines producing motion would also be considered motors (what is under the hood of a classical ICE car is both an engine and a motor, and is just a motor in other Germanic languages), and most motors producing motors would not be referred to as engines (an EV does not have "electric engines").
So for the types of devices where you are likely going to have the discussion - things makings stuff move - motor is the supertype and engine is the subtype.
While I think I agree with your definition of "engine", I'm not sure I agree with the idea of "motor" being a subtype of "engine".
To me, a "motor" is a specific part of a system's powertrain: it's the part which converts some other type of energy into kinetic energy for the purposes of moving the system. On the other hand, an "engine" is a self-contained system that establishes some kind of stable feedback cycle to perform a task.
With that in mind, a car's internal combustion engine is the motor of the car, since that's the place where chemical energy becomes kinetic energy, which results in motion after being transferred through the drivetrain.
If you take the same exact engine out of the car, mount it in some fixed position, and rig it up to generate electricity, I wouldn't technically call it a "motor" anymore, because it's not acting to move the system it's a part of. (That being said, I probably wouldn't split hairs here, since the engine was originally designed to be a motor. But you see my point.)
Importantly, this lets us fit diesel trains into our definition: they use diesel engines to generate electricity, but electric motors to actually make the train move. In an old steam-driven train,
It also addresses the question of what to call the part of a model rocket that makes it go up. It's the part of the rocket's powertrain which converts chemical energy into kinetic energy that moves the system. It's a *motor*.
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So, with all that said, an interesting point is that the terms "motor engine" and "engine motor" both make sense, but don't mean the same thing. "Motor engine" describes an engine designed to perform the role of motor (as opposed to a "generator engine"), whereas "engine motor" describes a specific kind of motor (as opposed to a "chemical motor").
I view "motor" as a subtype of "engine". An engine drives a process or system, either literally (producing power from some fuel or other energy source) or figuratively (as in Google's search engine). A motor is a type of engine that changes the momentum of some object.