`post_updated_messages` filter

The function you attach as a filter needs to take in a variable as an argument and return that variable once it’s done processing whatever it is you need it to do. For example:

function wpse426272_messages( $messages ) {

  $messages['game'] = array( 'Game updated', 'Game published', 'etc' );
      
  return $messages;
}

add_filter( 'post_updated_messages', 'wpse426272_messages' );