How can you tell if a mobile page has been specifically designed for mobile devices?

Let me express my understanding of your task: You want to check if one web page is, for example. https://www.google.com , is for both a mobile device and a desktop computer (this means that two types of visit can cause the server to provide a different web page) or just for a mobile device or just for desktop.

If my understanding is correct, then my answer is:

You can change the "user agent" in your HTTP request and then analyze if you can get the same or different content from the same URL. More about the "user agent" is explained here: https://www.thoughtco.com/detecting-hits-from-mobile-devices-3469093

After you get the content from the different "user agent" settings in your http requests, then if the content of the mobile device type "user agent" is different from the content type "user agent" desktop, you can assume that this page is the URL specially designed for mobile device.

But since there is no specific international standard to tell the client whether the content is mobile or desktop, so your job is to make that decision after you receive the same HTML content from both the user agent mobile device and from the "user agent" setting on the desktop.