Removing sensitive information from comment notifications – notify_post_author

There is a hook that should let you alter that data– comment_notification_text.

add_action(
  'comment_notification_text',
  function($notify_message,$comment_id) {
    var_dump($notify_message,$comment_id);
    die;
  },
  10,2
);

You could parse that $notify_message string and remove the parts you don’t want.

add_filter(
  'comment_notification_text',
  function($notify_message) {
    $notify_message = explode("\n",$notify_message);
    foreach ($notify_message as $k => $line) {
      $header = trim(substr($line,0,strpos($line,':')));
      switch ($header) {
        case 'E-mail':
        case 'URL' :
        case 'Whois':
          unset($notify_message[$k]);
        break;
        case 'Author' :
          $pat="([^(]+)\(.*$";
          $notify_message[$k] = trim(preg_replace('|'.$pat.'|','$1',$line));
        break;
      }
    }
    $notify_message = implode("\n",$notify_message);
    return $notify_message;
  }
);

I think that accomplishes what you were going for.