registering does not mean that you use a third party.
it means that,
you buy and record that a so called "domainname" is yours, so that everybody in world can reach you without any specific setting in their computers.
you have to do "domain registration" in one of "registrars" all around world, to make your domain reachable from all world.
After registration(buying) of domain from one of registrars, you "redirect" your domain to your server via "dns setting in registrar", then, you do hosting in your server without any third party.
You cannot see the registrar as a third party.
Domainname is a "name" that makes people remember something, that points to an ip of server.
in fact, it is an easy way of reaching a webserver/server.
if somebody wants to reach a server by name, there are a few ways:
* define a name-> ip redirection in local computer (such as hosts file)
* define name->ip redirection in the dns server of that computer/network
* define name->ip redirection in master dns servers or in one of "registrars"
the first two requires no registration or buying of something, in fact, it works, but everybody in world need to do that setting or define that dns server.
in third case, everybody in world is already using registrars and master dns servers, so, they dont need to do extra setting, so they can reach you.
putting your domain->ip pair/registration in master dns servers or registrars is called "registration" and it costs something.
if you dont want to register your domain, or if you dont want to pay something,
and if you want your domain to be reachable by people, then,
you need to say all people: "make that domain->ip redirection in your computer or your dns server". that is an option, but quite hard
third way, is how today's internet works.
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