Jesus Christ. In 25 years of working on the Web, this is the first time I'm hearing of "third level domains". WTF? Where and when did this come from?
No shade to the folks being harmed by this situation. Sorry. I'm just really non-plussed at the idea that the registrar is charging for what looks like should be a DNS prefix.
.name has been doing this for 25 years so I guess you must've missed them right when you got started!
For .uk, this scheme was the default until 2014 (a long time, considering it's first use was in 1985). Everything was .co.uk or .org.uk and you bought those directly with registrars. Then, in 2014, .uk registrations were allowed, and the .co.uk and .uk domains now coexist.
It’s so ad hoc that we need to maintain reference datasets to identify what’s registrable:
A real mess.
there’s nothing special about any level of domains, it was always a hierarchical protocol
You've never seen example.co.uk? Many countries use them because the top level domains for gov, edu, etc are restricted to one country.