Old large format camera had a front lens board, to which a flanged metal ring would be attached, and then the lens screwed to that ring. It looks like this:

Compare to a toilet flange:

… pretty much the same, right?

So, that's where the word flange comes from, even if it doesn't necessarily literally apply to the mounting rings of modern bayonet mounts which don't have quite the same obvious jutting-out flange.

But that background doesn't really matter for the purposes of your question. The flange distance is the depth from the ring on the camera body to the recording medium (lens or sensor). This is, for all practical purposes, the same as the distance from the mounting ring on the back of the lens when the lens is mounted, since these rings touch.

The rear element of the the lens can be recessed further in the lens body, or it can protrude slightly further. It depends.

Is it dependent on the shape of the lens mount? Well, yes, kind of, but since all lens mounts I know of use a fairly similar ring, that's kind of an academic point.

You call this "debatably the most important measurment in a camera", but, really, while one can debate anything, I don't think there's much of debate here at all — it's not that important or interesting except for some esoteric considerations. It affects lens design, but not "accuracy of focus". A system with a flange focal distance isn't going to be be more or less accurate in focus than one with some other measurement.