I have my website (mydomain.com) connected with a vps server and now I want to setup another server and use subdomain from main domain for hostname as well the nameservers (e.g. srv1.mydomain.com should point to new server, ns3.mydomain.com and ns4.mydomain.com new server nameservers). I have created both nameservers (ns3, ns4) and pointed to new server IP address on domain provider, and I think it does one part. But I am unsure about the other part. I tried to add dns record on older server for the host srv1.mydomain.com and pointed to new server IP but it did not worked, but it did not worked. When I queried the dns record of "srv1.mydomain.com" in a tool, it shows error "DNS record not found, reported by ns1.mydomain.com".
I am pretty sure, I am missing some crucial step. Any suggestion, resource or guide please?
Update: (Example Scenario)
I own a domain "example.com" and I want it to be used with two different servers. The "srv1.example.com" will point to a server with IP address "10.10.10.100" and should use these nameservers (ns1.example.com, ns2.example.com).
On the other hand, I want to connect "srv2.example.com" with second server "10.10.10.200" and it will serve under these nameservers (ns3.example.com, ns4.example.com).
The actual domain "example.com" needs to be hosted on first server "10.10.10.100".
PS: apologies for not sharing real domains, the company privacy doesn't allow that for privacy reasons.