Then what exactly is the lay-url for?

For providing a url to link the static text 'Lay summary'. The lay summary is not the cited source so does not have the same 'privileges' as |title= and |url=. An archaic hold-over from the olden days. Were it up to me, I would get rid of all of the lay parameters in favor of having editors write separate, complete citation templates (with author, date, work, and all of the rest) ...