When you want to destroy a session completely, you need to do more then just
session_destroy();
First, you should unset any session variables. Then you should destroy the session followed by closing the write of the session. This can be done by the following:
<?php
session_start();
unset($_SESSION);
session_destroy();
session_write_close();
header('Location: /');
die;
?>
The reason you want have a separate script for a logout is so that you do not accidently execute it on the page. So make a link to your logout script, then the header will redirect to the root of your site.
Edit:
You need to remove the () from your exit code near the top of your script. it should just be
exit;