There's nothing in that page to change except for replacing $action with $_POST['action']; all the problems must be in those ".inc" files. Where they mention $update_user, and whatever variables named after other form fields those includes mention should be changed similarly.
Putting extract($_POST) at the top of that page may be suggested, but it should only be regarded as a dirty patch designed to get it working while you take the time to do it properly.
And I'd point out that this isn't specifically a PHP5 issue: if this code is anything less than about four or five years old (depending on how you count it) then it was badly written in the first place 🙂.