It is indeed as my learned ATCO friends have said. No contention there.
Where it gets interesting is with older airfields that have changed "shape" over time, and hence the ARP has "moved". I had lots of fun with these when compiling the A4A airfields list! In the end, I decided that two ARPs less than 500m apart were probably the same place.
Similar with ex-RAF fields that have been returned to agriculture etc but where microlighting etc takes place on one part of the former field.
I have a suspicion that the RAF etc during WW2 may have been a bit less precise about how they defined the ARP - perhaps the CO's office?