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`

scenes/screens/lobby_screen/lobby.tscn

@@ -41,6 +41,6 @@ script = ExtResource("4_glu7g")
 [connection signal="pressed" from="ui/body/right_body/vbox/8ball" to="." method="_on_mode_bttn_pressed" flags=18]
 [connection signal="pressed" from="ui/body/right_body/vbox/9ball" to="." method="_on_mode_bttn_pressed" flags=18]
 [connection signal="pressed" from="ui/body/right_body/vbox/snooker" to="." method="_on_mode_bttn_pressed" flags=18]
-[connection signal="pressed" from="ui/body/right_body/vbox/free_ball" to="." method="_on_mode_bttn_pressed" flags=18]
+[connection signal="pressed" from="ui/body/right_body/vbox/freeball" to="." method="_on_mode_bttn_pressed" flags=18]
 
 [editable path="ui"]

scenes/screens/lobby_screen/lobby_base.gd

@@ -60,4 +60,4 @@ func _on_match_create_bttn_pressed() -> void:
 	MessageBus.emit_request.rpc("match_list_add_entry", match_entry)
 
 func _on_mode_bttn_pressed(button: Button) -> void:
-	active_mode = button.name.remove_chars("_").to_lower()
+	active_mode = button.name.to_lower()

scenes/ui_controls/lobby_screen/lobby_ui.tscn

@@ -158,7 +158,7 @@ mouse_default_cursor_shape = 2
 disabled = true
 text = "Snooker"
 
-[node name="free_ball" type="Button" parent="body/right_body/vbox"]
+[node name="freeball" type="Button" parent="body/right_body/vbox"]
 layout_mode = 2
 mouse_default_cursor_shape = 2
 text = "Free Ball"