Problem including jQuery

I’m having problems including my own jQuery on a page. Basically, I have an image map, and when regions are mousedover, I want to change the text contents of a div. This works fine outside WordPress.

I believe I’ve called the script correctly:

if (!is_admin()) add_action("wp_enqueue_scripts", "my_jquery_enqueue", 11);
function my_jquery_enqueue() {
   wp_deregister_script('jquery');
   wp_register_script('jquery', "http" . ($_SERVER['SERVER_PORT'] == 443 ? "s" : "") . "://www.mydomain.com/scripts/the-script.js", false, null);
   wp_enqueue_script('jquery');
}

and indeed it is loading, but it won’t execute.

The script is:

$(document).ready(function( $ ) {
		$("#ignition").mouseover(function() {
				$("#label-container").html("Ignition switch");
	});
	$("#ignition").mouseout(function() {
				$("#label-container").html("Point your mouse over the above picture for an explanation about what everything does!");
	});
});

Any reasons why this wouldn’t work? I’ve read various pages about not using $, and using jQuery, but I’m a little confused.

Any help would be appreciated.

Apologies I can’t link to page in questions, it’s on a closed site.