I would like to locally resolve the server names in my green network. Network, DHCP Server lists the local names of the servers correctly but I cannot ping them or typing ssh myserver results in an error message:
ssh: Could not resolve hostname myserver: nodename nor servname provided, or not known
is your systems DNS search domain equal to the domain of the server FQDNs? If not, you will need to provide the FQDN to ssh (which I recommend anyway, as it is more robust if your search domain changes).