jQuery AJAX http GET跨域 - 响应中需要的标头

Question

I've made a web application running on domain http://app.mydomain.tld and I've APIs on https://api.mydomain.tld (APIs are not developed by me).

I'm doing an AJAX HTTPs GET call with jQuery using this method:

$.ajax({
    url: "https://api.mydomain.tld/GetSomething/read.php",
    method: "GET",
    contentType:"application/json; charset=utf-8",
    dataType:"json",
    async:false,
    headers: {"Accept": "application/json; odata=verbose" },
    success: function (data) { doSomething(); },
    error: function (data) { showError(); }
});

This call is returning an error like NetworkError: failed to execute 'send' on 'XMLHttpRequest': Failed to load 'https://api.mydomain.tld/GetSomething/read.php'.

I've added the Access-Control-Allow-Origin extension for Chrome and the call is working fine.

So I've checked response headers. When Chrome extension is NOT enabled I've this response headers:

Access-Control-Allow-Origin: *

When the extension is enabled I've these headers:

Access-Control-Allow-Headers: access-control-allow-methods,access-control-allow-origin
Access-Control-Allow-Methods: GET, PUT, POST, DELETE, HEAD, OPTIONS
Access-Control-Allow-Origin: *

So my question, that could have an obvious answer, is: are these headers required to make correct cross domain calls? If yes, do my APIs need the below additional code?

<?php
header("Access-Control-Allow-Origin","*");
header("Access-Control-Allow-Methods","GET, PUT, POST, DELETE, HEAD, OPTIONS");
header("Access-Control-Allow-Headers","access-control-allow-methods,access-control-allow-origin");
?>

Is correct what I said or am I missing something to make my call working without changing APIs code?

PS: if possibile I would not use JSONP.