Hi, I’m wkk from Montréal.
I don’t understand where you want to get to with your post, but my two cents on what is a game engine:
A game is a technical challenge first. Your computer only moves data around, and a game is a program that emulates an interactive environment of some sort. Writing such a program is really complex.
Game engines are effectively programs, maybe frameworks, that implement all the basic components for making games, kickstarting your project by not re-implementing complicated systems (rendering, animation, sound, input, etc…). Depending on the quality of these pre-made components, your game can be more or less enjoyable (ulgy vs pretty, stable vs unstable, etc). Keep in mind that game engine users (the actual game developers) still have a responsibility to not implement the game mindlessly: you can easily misuse an optimized engine, resulting in an non-optimized/unstable game.
Then, on top of the core of these engines, usually some kind of editor is made (the graphical interface used by game designers). This allows people to work on the game without writing too much code or even defining level data by hand. Usually much easier to drag and drop things from some window/interface, although can be limiting in some scenarios.
I hope this helps you?