jQuery AJAX http GET cross domain - required headers in response - php

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.