It cannot really make a difference whether the stopwatch is started at 'Take-off clearance' or at 'start of roll', provided the same trigger point is used for both (all) aircraft. A crew's reaction time after receiving clearance is (slightly) variable of course - though with a couple of minutes notice everyone should be ready 'immediate' - the only one that is pretty fixed (+/- 5 seconds) is 'start of roll'. I was told years ago that ATC use 'Take-off clearance' for timing because they cannot stop 'doing' and wait to observe the start of every take-off roll. If ATC and crew use the same trigger, clearance for #2 is going to be issued in the last 15 - 10 seconds of the 2 minute period; if the crew observe that it is issued at about 1m30s then ATC are probably using 'clearance' (it doesn't matter, If you timed from start of his roll then you start to roll at 2 minutes).
Rotate point is a variable along the runway, even for the same type of aircraft, and a LIGHT will usually be airborne before the preceding MEDIUM/HEAVY so no issue there.
On the 'Rotate' issue, the wing generates lift as soon as it presents a positive angle of attack to the airflow, this lift force increases progressively until (at the appropriate speed and AoA) it generates sufficient lift to carry the aircraft mass. So lift is generated very shortly after the nosewheel unsticks - but not much, initially.