Minor refactorings.

- Move generateId to RoomManager since it's the only thing responsible
  for generating the five-letter IDs.
- generateId now makes lowercase-only IDs.
- WebSocket type alias exists only in GameManager now.
- Move GameType back to its own file, since no more packages needed.
GameManager refactor.

- Document the types param of the default impl.
- Move the WebSocket type to the trait.
Merge branch 'generateId'