You can't just copy and paste the mysqli usage. That was done as an example. You need to figure out how to make it work with your custom $site->db object.
As of PHP 5.5, the MySQL library has been officially deprecated. It is recommended to move to either MySQLi or PDO libraries for your mysql connectivity. See here for help choosing which interface you prefer: http://php.net/manual/en/mysqlinfo.api.choosing.php