fix: refine game mode state tracking and turn handling

* Centralize ball state generation in `BaseMode`
* Track turn count, player, and ball state consistently
* Prevent 8-ball, 9-ball, and colored snooker balls from being marked valid when reaped/sunk
* Update mode initialization and ball-state assignment
* Normalize Free Ball mode node name to `freeball`

controllers/game/modes/game_manager_base.gd

@@ -21,5 +21,5 @@ func _balls_stopped_moving() -> void:
 	mode.handle_switch_user()
 
 func set_mode(game_mode) -> void:
-	mode = Globals.game_data.GameMode.get(game_mode).new()
-	mode.generate_ball_state()
+	mode            = Globals.game_data.GameMode.get(game_mode).new()
+	mode.ball_state = mode.generate_ball_state()