You simply don't send "submit" with the request. You could add it here:
$(".messageForm").load("inculdes/signup.script.php", {
firstName: firstNameInput,
lastName: lastNameInput,
submit: 1
});
or check for isset($_POST['firstName'])
Sidenote: If there wasn't an object passed in with load()
it would be a GET request. Worth remembering...
Request Method
The POST method is used if data is provided as an object; otherwise, GET is assumed.