I decided to learn the game a little by doing single levels runs. What is the way community measures time? Am I suppose to use live-split or should I use in-game timer? (for example, physical training gives you 2 min at the beginning, I finish at 1m 22, so my time is 38 seconds?)

Yeah in-game time is preferred, only Physical training needs you to calculate the true in-game time, other levels should either be in-game time if shown at the end or real time.