Think more like "boxing ring" in that everything inside the circle is "in the ring".

The weird one is actually "rink" as in "skating rink" – because it derives from the word for "line".

So what you're talking about might actually be called a "web rink" if it existed, in that links flow in a linear fashion.

No, the original webrings from 1995 were literally rings - each site linked to one other site in the ring, and eventually one of them would link back to the “first” site, making an actual ring.