We signed up for our first OneWorld rtw and learned that our European itinerary counted as five segments:
(BKK)->LHR->CPH
CPH-LHR-LIS
LIS-MAD-ATH
ATH-LHR
LHR-(JFK)

Stopovers are CPH, LIS, ATH and LHR (just before JFK). The other intermediate cities shown are normal 2-hour connections, approximately.

I'd think that would be either six segments (if they count the intermediate cities) or four (if they don't). Sage words (better, suggestions on how to avoid the extra-segment charge) would be much appreciated.