How to change domain name?

If you asking about using a different domain name with a current server, just point the DNS to your server and point the old one away from it. The server name is set at provisioning and in line with the vendor.

To see your hostname, use hostname in the terminal. To change it, type sudo hostname new-name. The new host name won't stick with a restart. For that, you also need to go to sudo vi /etc/hostname and change it there as well.

If you are trying to rename the site on a forge server, I think that is possible in the meta tab of the site. You should probably redeploy the site after changing the name in that panel. I could see how problems could occur because of the way forge works.

I own a domain that I created on Forge. From now on I will be using another domain for my server. This is my case and I don't want to do it by redeploying :) I would have been great if I could just rename in forge, change NS in my Vultr account and ready to go

You can do it as I said but the docs do not say whether you must redeploy to the newly created site. SSH into the server and copy the current site directory under a different name. Then try the rename option in the meta tab. This way you can copy back the files if something goes wrong. Just an option among many.