If you make this change you are just updating the trigger to be 'same page post' to be 'any page post'. This will make the page send the email whenever a page posts info to it. You should add in another condition to this if statement to check for the register button being pressed, it might look like this:
<?php
if (($_SERVER["REQUEST_METHOD"] == "POST") && (isset($_POST['Register'])?$_POST['Register']:"" != "")) {