I run my own nameserver (with HE as a slave). I set up the nameserver, and I registered its address as a nameserver on Namecheap, but I'm not entirely sure what this does. From related questions I've deduced it adds a glue record to the root nameservers, but is that all it does or does it do something more?
You only need a glue record if your DNS servers exist inside of the zone that they are authoritative for.
Registering your DNS servers with your registrar usually registers NS records that point to your DNS servers so that they are then authoritative for that zone.
Of course, if you have any question about what your specific registrar is actually doing, you should call their customer support.