I've been looking around and I'm struggling to discover if a domain server is different than a domain name server, which I understand to be a server that maps URL names to IP addresses.
Are these two things different? Is it a different shorthand for DNS?
Domain server AKA domain controller is a server computer that responds to security authentication requests (logging in, checking permissions, etc.) within a Windows domain.
A domain name server (DNS) maps domains names to IP addresses, in a nutshell, as you said.
They are not related, not a shorthand.