Sorry if this is the wrong section, I read the pinned post and I think it's right but tell me if not.
Background: I've been building a small site for about four months, it's nothing dramatic, but I don't want it connected to my name. Not because of anything illegal — I have a normal job and I'd rather my employer not find it if they went looking, and I'd rather the people who might not like the site not be able to find my address.
I've read that WHOIS privacy is free at most registrars now so I assumed that solved it. Then I read that it doesn't. Then I read about proxy registrants and got confused about what the difference actually is.
So my question is basically: is "a domain nobody can trace to me" a real thing I can buy, or is it a thing people say? Thanks in advance to anyone who answers, I know this is basic.