This is interesting. Below is a crazy suggestion. Remember, your question is also equally crazy.
Your website, once opened through a browser, should generate a long polling connection (Comet programing). This will create a unique session between the browser and the server. When ur JS is making the ajax call, send some token (unique token every time) to the server through the long polling thread. Let the AJAX also send the same token. At the server, get the AJAX token and check whether you have a similar token in you long polling session. If yes, fulfill the request. Any coder can break this. But, it won't be easy. Chances are the freeboarders won't even see these second piece of comet code. You can implement the comet code in such a way it is not easy to detect or understand. When they call ur service, send a 'Service Unavailable' message. They will be confused. Also make the comet code https.
You can also check how long that long polling thread is open. If the session was just opened and you get a ajax call right away, you can assume it is a 3rd party call. It depends on ur website flow. If ur Ajax call happens after 1 second of page load, you can check for that pattern on server side.
Anyone coding for your public api, will have 1 to 2 secret checks that they wouldn't even know and even if they know, they might be discouraged by all the extra coding they have to do.