I use a simpler categorisation:
S Wales (any cave in S Wales)
FoD (any cave in Fod)
Devon (ditto for Devon)
Mendip (ditto for here)
Derbyshire (any cave near Derbyshire)
Yorkshire (any cave near Yorkshire)
(Not in any particular order BTW).
Then, when describing somewhere more specific to someone, you can define a more accurate area by pin-pointing the site with relevance to somewhere else; i.e. on Mendip we might use Northern Mendip, Southern Flank, East or West or define a general location and then add other location names to pin-point a smaller area such as Ebbor Gorge (near Wookey), Burrington Combe, Priddy, Leigh-on-Mendip, Rookham or Cheddar Gorge. I imagine each caving region will have a similar defining method which will have more relevance to people living in that region with a better knowledge of micro-regional locations (which will not be much help to an outsider unfamiliar with the smaller locations) and so each region will have spokespeople who will highlight their own peculiar regional versions of the above outline for Mendip.
Speaking as a southerner who hasn't a clue about the geography of northern England, the generic term "Yorkshire" is very helpful to me; never having knowingly been to Sheffield, Cumbria or Lancashire these names are of no help to me since I don't actually know where they are but I do know that Yorkshire is a long drive to a cold land where there are very friendly people who drink in pubs which never shut and the caves are ace and the underground streams are freezing cold even in summer.