Solution 1:

You can rename an AD domain with rendom.exe. There are plenty of caveats that you should be aware of when doing this. Read carefully.

That said, it has nothing at all to do with what you actually want to accomplish. Your best bet is just configuring VPN access to your internal domain.

Also, as another side note, never use .local or any other made up TLD for your AD Domain's TLD

Solution 2:

"I now would like to be able to access the server from outside my local network"

If that's all you wanted to do, you only needed a VPN setup. No external domain name needed to be purchased, AD authentication would work just fine, and you (assumption here) wouldn't have needed to open up your IP of your server completely to the outside world (sure hope that isn't what you did).

If you are wanting to access the server over the internet, use a VPN (set it to your external WAN IP). If you really want something fancy like "vpn.domain.com" then setup an A record on your newly bought domain for that and set it to your WAN IP, but still setup the VPN correctly.