A game development library, is a library intended to be used by game developers as a part of their game.
A game development framework, is also a library, which also covers most of the needs of game development and provides extension points for the game specific code to be added.
A game engine is at least a few components:
- An executable runtime, which is what actually make the games run.
- An authoring tool (game development environment or editor) which is used to create games.
- A building tool, which converts the projects made with the authoring tool into the final game (which uses the executable runtime).
I'd argue that what makes engine the engine is the executable runtime. Addendum: I'm not trying to define what features you would expect an engine to have, but what is the bare minimum to make the category.
One possible way to work around this is to create the games as a separate package (including scripts and other assets), which the runtime loads. This way the package can be under a license unrelated to the runtime. Addendum: While I frame this is a workaround for copyleft, this is how many non-copyleft game engines work, both open source and closed source.
It is also worth noting that LGPL might allow developers to use the code as a library... Which means that to allow game developers to use any license on their game, you might make your code as a library under LGPL that game developers can load it oninclude with their code... Meaning that it would be closer to a game development framework than to an game engine.
Be aware that this answer is not a replacement to reading the aforementioned licenses or asking a lawyer.
Addendum on adding an exception to GPL: I agree that parts of the code can be exempt from the license. However, the way I read the question the exception would have to be based on usage (using it for creating game vs using it for creating game engines). I want to reiterate that I'm not a lawyer to emphasize that I do know if such exception would be compatible with GPL. Please hire a lawyer instead of relying on comments on the internet alleging it to be possible or not.