My personal record for a departure being recorded as "on time" was one that pushed 50 minutes after STD! That was the airlines decision.
It depends on the airline, the ground handler, and what the actual reason for the delay is. Generally up to 5 minutes late and we can usually call it "on time", anything beyond that requires a delay code and time to be recorded to explain why the flight has departed late.
I tend to go by actual pushback time, but you can use the brakes released time (which is what ACARS uses to catch us out when we try and hide a delay!) or the "a/c flashing" time. As you've noticed, there are no hard and fast rules!
I believe the official CAA punctuality statistics count anything departing within 15 minutes of STD as an on time departure.