It should be showing in that Server Behaviors window but it isn't, which can happen if there's any manual code added to the server behavior, or sometimes if there are any blank lines in the PHP code it won't display. I've copied your code into a page here but because it's an old version of Universal Email, I can't get it to show in the panel either. Can you please let us know which version of PHP you're running though? If you upload the attached file to your site and then load it in the browser, it will tell you - this info is crucial before going any further.
I think your best bet is to install the latest version of DataBridge and then go to WebAssist > Send Email and add a new server behavior for the existing form, then remove the old code. You will then see the behavior in the panel and you can add the information you need.
One more thing that's crossed my mind - is the form submitting OK but not sending the email, or is it failing validation? If it's submitting OK but not sending, add the following code to the top of your success page (ESuccess.php) and when you submit a message, it should display any errors at the top of the page: