After finishing a conversation and hanging up, the phones will (sometimes) then ring 3 times. I haven't made particular note if it is both after I received a call or made the call myself, but definitely the former, and from someone I know.
Don't have another phone (3x BT Aura 1500)
Recently had a fault whereby we had no dial tone but broadband was fine. Engineer visited and fault traced to exchange. He also upgraded/replaced the phone socket and added a filtered faceplate. No wired extensions etc, with router & phone direct into master socket.
We have had the issue before, but I can't remember what/if any particular event cleared it up, but it hasn't done it for a long time, till now.
It could still be an exchange fault, but you need to check at the test socket which will be located behind the filtered faceplate.
What does your master phone socket look like? It should be 5,6 or 7.