I saw somewhere on the docs that diskless clients on the main LAN should work:
"Interestingly, you can choose to run a diskless workstation or a thin client on the skolelinux 10.0.0.0/8
So, I tested again client1 (the one on the LTSP LAN) and it was having the same problems as client2 (the one on the main LAN). But it was working a few days ago. I remembered (after trying a few other things) that a few days ago I was giving 2GB RAM to the diskless client, and later I switched to 1GB RAM. I removed the client and created it again with 2GB RAM, and it was working again. I was lucky.
This seems a bit strange because the Xfce desktop (that I am using) should be fine with 1GB RAM.
It brings the question what are the minimum requirements for DebianEdu (at least just for testing).
Currently I am using 4GB RAM for the main-server, and 2GB RAM for the diskless clients.
About your suggestion: I didn't do any configuration on the server for the diskless clients, and I was actually expecting it to work, because I saw somewhere on the docs (at GettingStarted) that diskless clients don't need any configuration.