"and the name server accepts queries from the internet at large, a designation known as recursive"
A DNS server is not recursive if it accepts queries from the internet.
It's recursive if it tries to resolve queries for domains it's not authoratative for, by passing those queries on to the actual authoratative servers (those listed in the NS records for the domain) and then passes the reply back to the client.
So if your DNS server is the NS server for "theregister.co.uk", then it will resolve things like www.theregister.co.uk to an IP, if queried.
If it's recursive, and you ask it for the IP of www.google.com, your DNS server will query googles DNS servers, and return the IP it gets back to you. If it's not recursive and you ask for www.google.com, it will respond that it has no record of that name.
It's got nothing to do with accepting queries from the internet, a DNS server on a LAN can be recursive or not, depending on what you need to achieve.