No it sure isn't working. They've managed to do something to rewrite it. DOCUMENT_ROOT should be virtually mapped to you, but it clearly isn't:
The DOCUMENT_ROOT should be pathed to /home/mysitename/public_html, but it has been improperly configured from the looks of it. Best to contact your host about that.
Do the relative to this directory approach. Overall its far more useful anyways since you can use it in cron jobs while DOCUMENT_ROOT cannot be (unless you manually set it).
require_once __DIR__ . '/../includes/header.php';
The only thing to remember is you will now always go relative to THIS file that the require is called in, not to the file that is currently executing. This does make nested includes a lot easier to work with though since you never need to concern yourself with where it is running from.