Sorry, I typed it quickly and had the slashes backwards… Getting old, memory going… LOL
So, two things… You did save the file as a PHP file not HTML, correct? Also, did you fix the < FORM > tag to point to itself and you might need to alter the PHP code to check for the submit button. Do you know how to DEBUG this? You can check the inputs from the posted form if needed. Something like this would work:
<?PHP
die("<pre>".print_r($_POST,1)."</pre>");
?>
This should show you all of the posted fields from your < form >. If there is data in there, you can see what was actually sent to you and adjust your code as needed. Or, you can create a test file and post it here and we can help you solve where it is failing. But, it appears most of it is working. It seems you are just not getting the data sent back from the form.