The problem in question was that the search result displayed in full screen, instead of inside the IFrame, as supposed.

The solution was as simple as adding the following line (bold below) in the results part of the code:

<script type="text/javascript">
        var googleSearchIframeName = "cse-search-results";
        var googleSearchFormName = "cse-search-box";
        var googleSearchFrameWidth = 100;
        var googleSearchDomain = "www.google.com";
        var googleSearchPath = "/cse";
        var googleSearchResizeIframe = false;
    </script>

If you struggle with the same problem, give it a try. It worked for me.