I found the theoretical discussion in this post very interesting but at some point it has to be brought back to a practical application in flight training.
The POH numbers are a guide and only apply to the exact conditions present during the test flight (and a new airplane and consistantly correct control inputs throughout the test). So assuming we are starting with a 30 plus yr old, hard used rental, and a pilot who may be safe but just a little bit rusty and you have allready invalidated the assumptions in the POH. Plus of course you have the imponderables like the fact the first half of the field may be nice and firm but the second half isn't as well drained and therefore is very soggy.
I tell my PPL students to calculate from T/O distance from the POH and then add a 50% additional safety factor. I also teach then to be wary of the takeoff trap of thinking that that even though the end of the runway is approaching I am "almost" at takeoff speed and so if I continue it will work out. I like the 2/3 of flying speed at 1/3 distance as a working rule of thumb as it is easy to calculate. I also make sure they pick a definite visible marker at the 1/3 point alongside or on runway as estimating the 1/3 rd point during the actual takeoff is not really practicable. Finaly I emphasize the importance that the aircraft must be accelerating throughout the takeoff. If the airspeed stops increasing than the takeoff must be aborted.