nameserver - Is it required to register a name server with GLUE records?

I have a few domains and want to resolve their DNS records with my own name server.
Let's say I have a DNS server with 2 fixed IP addresses and a domain name mydnsservers.net.

I'd like to have 2 name server -subdomains- for my other domains.

• ns1.mydnsservers.net > 81.250.18.12


• ns2.mydnsservers.net > 81.250.18.13

Can I just use a third party DNS (e.g. AWS Route 53) for mydnsservers.net and setup two A-records like this?

ns1. A 81.250.18.12
ns2. A 81.250.18.13

Or is it mandatory to use my own DNS server for mydnsservers.net and configure GLUE records at the TLD registry?

I know that the first option works in some cases, but my new registry gives an error when trying to use ns1.mydnsservers.net for one of the domains because it's not registred as a nameserver (doesn't have glue records).

Any help would be much appreciated!