No you don't have to register the www.mydomain.com
by your registrar !
A 'A' record in your mydomain.com zone represents a host (If I am right). A CNAME record represents an alias for a host.
If you want to create a real subdomain, I think you'd better try to create a new zone.
But here if you just want your dns to resolve the host www.mydomain.com,
insert a A record, don't forget to increment the serial and perform a killall -HUP named
I run such a configuration on my lan here at home (only with hosts) and it works fine (the domain is a fake one and is not reachable from the internet but when you're in the lan it works).