when you use data-request on form element, then payload of that request will include all inputs, same, when you use data-track-input then data-request-query is not needed. on ajax handler check for post(‘search’) or \Input::get(‘search’);
We need data-request-query to work anonymously. So it will do as @snipi says and place all the form data in the query string (similar to the behavior of <form method="GET">)