A name is a dynamic thing, and servers especially can have more than one. Take for instance a .NET web-app server such as used for Outlook Web Access. Depending on the roles it has, it can have any of the following names:
- The AD name of the server itself (Exch2010-CA1.AD.example.com)
- The internal DNS name of the OWA server used for convenience (webmail.ad.example.com)
- The external DNS name of the OWA server used for outside logins (webmail.example.com)
- The external DNS name in the MX record for example.com (smtp1.example.com)
DNS names are not one-to-one with servers. Servers in shared-hosting farms can have thousands of DNS names pointing to the same server.
As for the active directory DNS-domain, in my above example, the DNS domain 'ad.example.com' may be completely unresolvable from the outside. Intentionally so, even. It prevents those tasty SRV records being used as a hacking hit-list.
In your case, DC01 should probably have a private address and not bother with a public one. Remote access to that would be done via public IP, probably proxied by way of a NAT gateway with a port-forward, and the DNS used there would be whatever you want it to be.