If I host my website on Digitalocean and purchase my domain from namecheap.com, I have to edit the Nameservers linked to my domain (www.example.com) in my namecheap.com profile with ns1.digitalocean.com, ns2.digialocean.com, ns3.digitalocean.com.
Does that mean when the Recursive DNS server queries for the IP address from the Authoritative nameserver (namecheap.com servers), the namecheap.com servers will act as a recursive DNS server and redirect the query towards Digitalocean?
namecheap.com is a Domain Name registrant. It provides default name server for the domains bought from namecheap.com.
If you change the nameservers on domain name registrant, the DNS query from recursive nameserver will directly query the nameservers on digital oceans. There is no namecheap.com to digitalocean.com DNS query.