> In terms of "string that I can type in the Browser address bar to go to a website", I agree with your definition.
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.