I'm going to suggest a slightly different approach, as it sounds like you're not having much luck with the current one. What I'd suggest you do to keep things simple here, if I'm understanding your problem right, is to stop thinking about the redirect domain as a Moodle problem and just solve it as a website problem.
It sounds like you've already got a functioning Moodle site at www.mydomain.org.
What you want to be able to do is to redirect anything that hits mydomain.org/whatever to www.mydomain.org/whatever.
The site doing the redirect at mydomain.org shouldn't particularly need to know that the place it's redirecting to is being served by Moodle, and the Moodle site shouldn't care that when somebody shows up at its site, www.mydomain.org, they came via a redirect from a site at mydomain.org.
So get rid of the "Alias /mydomain/moodle /var/www/moodle23/web" business, set up a new, separate virtual host to serve mydomain.org with its own document root, and make sure you can serve a simple HTML page from it. Also make sure you didn't break your www.mydomain.org domain in the process and that domain still visible and serving Moodle. Your host should be able to help you with this if you can't figure it out.
Once you've got a functioning, separate site at mydomain.org, set it up to redirect. You should be able to do this either with the .htaccess provided upthread. If you're not allowed to use .htaccess, you should be able to put the same contents directly in the server config. This is another bit the host should be able to help with, as it's a very simple, common problem that doesn't require them to know anything about Moodle, but feel free to post again if they can't help you with it.