That isn't a domain either. An address in there is a URL, a partial or complete one (it also recognizes search terms, but that's different).
It's actually really simple.
> In the Domain Name System (DNS) hierarchy, a subdomain is a domain that is a part of another (main) domain.
Having a subdomain be a type of domain is really important for speaking concisely about web security. If I say "what's the domain of the cookie?" and the answer can't be "news.ycombinator.com", that's suboptimal. At the same time for security it is also important to know who the registrar is, but for that there is the term TLD.
In the ownership context the definition differs. Please if you want to challenge my notion of "subdomain" please only challenge this definition only. You don't own a subdomain as strongly as owning a root-level domain, because in the first case the only attestation is on the database of obl.ong/github.io/etc while in second case your ownership is attested by the ICANN registry, which is how things has been working for the past 20/30 years.
In terms of "string that I can type in the Browser address bar to go to a website", I agree with your definition.