Yes, apparently "SLD" is quite a common term but it does not encapsulate the likes of 'pvt.k12.ma.us' or 'uk.com' (in the sense that it's not a registry per se for the latter)
So there's an awkward way of defining things in various contexts, like cookie permissions, whether a domain is registerable, whether it is a registry the next level up
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?
Well, exactly. Canonically they're often referred to as TLDs in the context of "this trusted authority delegates sub-level hosts" but it means something different to many clients.
As the original comment suggested it's a bit of a non sequitur to call anything below the initial host label to be a TLD. I was just suggesting there isn't a generally accepted moniker for what they're called.
The public suffix list makes some kind of definition, mainly for cookie-level permissions.
