There’s no need to change our the login names for the new domain name. The name which is displayed there is only so that you can read it. SQL Server actually tracks logins by the SID which it gets from Active Directory.
As long as you are either doing a domain rename, or you are using the Account Migration tool (if you aren’t just renaming the AD domain) you shouldn’t need to do anything as all the SIDs will match already.
If you are building a new domain, and you aren’t going to migrate the users with the same SIDs then you’ll need to create new logins and new users as the SIDs between the logins and the users won’t match. You’ll need to script out the existing users so that you capture database role membership and permissions then delete them and recreate them using the new login names.