I think the sector referred to as 15 minutes is for revalidation/licensing purposes:
From LASORS:
Sector: A flight comprising take-off, departure, cruise of not less than 15 minutes, arrival, approach and landing phases
To renew a ME rating requires a 'sector', but I guess a 3 minute sector wouldn't really be of much use, hence the 15 minute bit.
I used to fly JER - SOU in TPs, and the cruise was often only 5 mins by the time we'd got to FL180 (on a 40 min flight), but it was still a sector.
In normal day to day stuff, a 'sector' is a flight. LGW - PMI is a sector. So is LGW - LGW (return for tech problem). If you are paid per sector, its the bit between each T/O and landing.