One you missed: The Self-Improver

The key motivation of the self-improver is to level up *themselves*.
It's got to be difficult - games that are not actually a challenge are just not interesting, somehow.
But a variety of different types of games, that provide challenges in different directions, and different things to be learned, can be interesting.

Actually, playing just one type of game will often lead to diminishing returns for the self-improver.
Switching to new and different games can often then be a quicker way to interesting and meaningfull new things to learn.

Some classic games for the self-improver are Rock Band 3 with pro instruments (play a game whilst learning musical instruments),
and programming games like Spacechem.