Do I need a second domain to run my own authoritative dns server?

I have a static ip (lets say 142.251.208.110).

I own the domain: website.tld

My registrar is godaddy.

If I want to change my nameserver godaddy won't allow me to enter a static ip. It wants a hostname. I observed that many use ns1.website.tld and ns2.website.tld.

I don't understand how this can work because ns1.website.tld would be served by my dns server which is not yet known by others.

Do I need a second domain like domains.tld where I use the registrars dns server for serving ns1.domains.tld which I can then use as the nameserver for website.tld?

I would like to avoid the registrars nameserver and avoid getting a second domain just for dns.

Thank you for your input.