Of course it’s self hosting. The term “self hosting” just means being in control of the service or host yourself, as opposed to that being controlled by a third party.
It doesn’t mean the hardware has to be in your house. It just so happens that that is the majority preference, because people value privacy, and are often hosting private data.
Personally, I have firstname@secondname.email That was a domain I set up when I was self employed, contracting for people, trying to look ‘professional’ I still use it to this day, though the professional requirement is gone now since I’m employed by a company. Still, it’s catchy and not that expensive, so I keep it, and renew for long periods of time.
Then I have another domain, not personally identifiable, for whatever stuff I want to host/play with. This one is a three letter .ac domain, I got 10 years for $200 ish. The three letters are my initials
So you say, is there usefulness in having a real name domain? Not unless it’s your business. For me, when I was contracting, it was a simple attempt to get clients to remember me over the hundreds of others
To me, ones like joeblacktechnicalservices@gmail.com always seem like a lazy attempt, for a ‘business’