I have a Server 2012 DNS server on subnet 192.168.7.0 and I am trying to use that on a host on the 192.168.8.0 subnet, which is not on the domain (this may not be relevant).
When I configure the 192.168.8.x host to only use the 192.168.7.10 DNS server, I can find hosts in nslookup by searching for computer1.mydomain.local but I can't find them just by searching for computer1.
Why do I have to add mydomain.local in the nslookup or ping?
Probably because you don't have any domain suffixes appended.
It's something you configure on the client, which doesn't magically know to append a DNS suffix to a hostname, unless you tell it to.
You answered your own question here. If the PC is not on your domain, it will not append that domain name to DNS queries. As a test, do a ipconfig /all and look for DNS Suffix Search List on that PC. If you don't see anything there, your PC isn't going to append any DNS suffixes to queries. Now do the same test from a PC on the domain and compare. You should see mydomain.local in the list on the PC that is on your domain.
