show something only when user comes from specific page at remote host?

$_SERVER['HTTP_REFERER'] gets all the info you need, not just the domain – so you just need to drill deeper into the data you are already getting.

I’d suggest only setting one variable to make sure you don’t paste something wrong into the second – you can check for the www version by appending ‘www.’.

Your current code would display the page to anyone who accesses the page directly with no referrer (i.e. by bookmark or search engine result) and it sounded like you only want people who successfully completed the quiz to be able to view the page, so I also removed the part where you set the domain to the right one if the referer was empty.

You may need to tweak this a bit, but it should point you in the right direction:

<?php
// set the non-www allowed domain, path, and title
$allowedDomain = 'thequizhost.com';
$allowedPath="/the-quizzes/quizzing.php"
$allowedQuery = 'our-specific-quiz-we-need-to-echek-for'
// grab the HTTP referer using WP's built-in function
$referer = wp_get_referer();
// if referer is not empty
if (!empty($referer)) {
    // split url to check its pieces
    $refererData = parse_url($referer);
    // check domain, both non-www and www
    if($refererData['host'] == $allowedUrl || $refererData['host'] == 'www.'.$allowedUrl) {
        // check the path
        if($refererData['path'] == $allowedPath) {
            // parse the query string
            parse_str($refererData['query'], $queryString);
            // check just the "title" variable
            if($queryString['title'] == $allowedQuery) {
                // now we know they really completed the quiz
                $showForm = true;
            }
        }
    }
}
// if $showForm is true, they've been verified and can see the form
if($showForm == true) {
    do_shortcode("[signup_form id='2']");
// they didn't come from the right place, so don't show the form
} else {
    // redirect to homepage, or wherever you like
    wp_safe_redirect(home_url()); exit;
}
?>

If you think there’s a good chance visitors will try to hit the page without taking the quiz, instead of redirecting, you could display alternate content – something to the effect of “Thanks for your interest. Please take the quiz (link here) to enable signup.”