Which sounds like the best job for you?
AI Programming seems really interesting. Sometimes seeing AI in games work incorrectly really gets you wondering how they're intended to work and what went wrong.
Core engine programmer sounds like too much work and seems complicated.
Which sounds like the weirdest?
Tools programmer. Making tools to help build a game inside a game engine with its own existing tools seems funny.
How is game design different from game development?
I would think that game development deals with the production and creation of game assets and things necessary for the game to run. Game design would instead focus on the choice making that is responsible for why games are the way they are. If we were to look at game design and game development in terms of 3D models, game development is the act of creating and molding a model. Game design encompasses everything else involved with the 3D models like style, what the models represent, their purpose, etc.