You'd think it would be number of laps but I usually just keep lapping the tracks and don't restart and the number of attempts is always lower than the number of laps I've completed, which is why I think it might be the number of improvements made.
Yeah I can tell your talking drifts not time trials ;-) Drifts I can see working off the number of times you finish a run.
Yea I did a lot of drifting last night and was really focused on how many attempts I made. That one drift is still infuriating that I couldn't get a better score...anyway. Interesting that attempts might update if your time improves. Will keep an eye on that.