If I am understanding you correctly, you are looking to correlate your requests and responses via a unique value. 
You could use context on jquery.ajax. This means you would have to "give up" load but since its really a wrapper around ajax you aren't losing anything besides a little typing.
Here is an example using similar code that you provided.
$(function () {
// Ensure if our request context doesn't have an ID we at least get something back.
function getContextRequestID(settings, defaultIfNotSet)
{
    if (typeof settings.context.requestID !== "undefined")
    {
        return settings.context.requestID;
    }
    return defaultIfNotSet;
}
function writeToJsResult(method, settings)
{
    var DefaultRequestID = -1;
    var requestID = getContextRequestID(settings, DefaultRequestID);
    $(".js-result").append('<p>' + method + ': Ajax request for page -->  ' + settings.url + ' with ID of: ' + settings.context.requestID + '</p>');
}
$(document).ajaxSend(function (event, jqxhr, settings) {
    writeToJsResult("Starting", settings);
});
$(document).ajaxError(function (event, req, settings) {
    writeToJsResult("Error", settings);
});
$(document).ajaxComplete(function (event, req, settings) {
    writeToJsResult("Complete", settings);
});
/* triggering all Ajax requests */
$('.js-btn-json').on('click', function () {
    loadBBC();
    loadBBC();
});
function loadBBC()
{
    loadSite("http://www.bbc.com");
}
function loadCNN()
{
    loadSite("http://www.cnn.com");
}
function loadSite(url)
{
    // Not sure how you want to correlate them. For now using random. 
    // Perhaps a global running counter
    // Perhaps a GUID/UUID
    // http://stackoverflow.com/questions/105034/create-guid-uuid-in-javascript
    var requestID = getRandomInt(0, 100);
    $.ajax({
        url: url,
        context: {
            requestID: requestID
        }
    });
}
function getRandomInt(min, max) {
    return Math.floor(Math.random() * (max - min + 1)) + min;
}
});