uk.com is second level, pvt.k12.ma.us fourth. I think you really look words for denoting political separation (instead technical), but I don't know any good terminology for that. Public suffix is probably best term to differentiate between "shared" and "dedicated" namespace.
Technically though, root servers have information only about TLD-s. co.uk may be special for some people, but a.root-servers.net. don't care, it's still sends you to uk TLD nameservers, like for every other second level domain.
That political separation or what counts as "registry" is itself very tricky. Are dynamic DNS services registries? How about hosting (like blog) service provider, who gives subdomain/hostname for every client?