Web PKI is just its name‡, like my name is tialaramex, names are not meanings. The Web PKI is a PKI for TLS services with names from the public Internet not just for HTTPS (the web).
Names meaning mostly DNS names like news.ycombinator.com but also numeric names such as 1.1.1.1
The Baseline Requirements (the rules Certificate Authorities agreed to obey some years back now) currently provide 13 potential ways to verify domain ownership, but these ways are sometimes called the Ten Blessed Methods dating back to when Gervase Markham was alive and worked on the Web PKI.
Although some of the Ten Blessed Methods are specific to the web in practice, such as 3.2.2.4.6 which involves creating a document to be retrieved from a web site, others really aren't such as 3.2.2.4.7 which is just a DNS change. Let's Encrypt offers both these (as challenges http-01 and dns-01 respectively)
Method 3.2.2.4.12 is specifically for domain registrars in the sort of relationship you're talking about, and is used by some registrars which also have a CA business (such as Google) today I believe.
‡ It got this name because the Web is why this PKI exists. SSL was invented by the Netscape Corporation, whose Web browser is the ancestor of today's Firefox. Netscape's successor, the Mozilla Foundation and Mozilla Corporation continues to be the most visible and only public oversight for the Web PKI. Perhaps if instead some other group had secured the Network it would be named, say, the Usenet PKI, or the IRC PKI or the NetBSD PKI, but they didn't and it isn't.