Some thoughts, if I may ...
Why do we have take-off segments?
Repeatability. By prescribing some sort of how to do it from A to B in a very simple, mechanical way, we can define technique, measure progress and deviation from the plan, etc.
Otherwise it would be a bit of a shambles and not the sort of thing we look for in heavy iron operations.
first segment
the takeoff involves getting off the ground and then selecting gear up (iaw the AFM - might be, for instance, two seconds after liftoff - whatever). Depending on the performance on the day and gear sequencing, the gear will be tucked up and away after a period of seconds at which stage the aircraft will be at whatever height ....
Should this process finish prior to the end of the TODR, there is no first segment and we start with the second segment from the end of the TODR (ie screen) - typically hot, high, heavy ie poorer performance.
Should this process finish after the end of the TODR, the distance between end of TODR and gear eventually up is the first segment - typically cold, low, light ie better performance
Some aircraft have a retract cycle sufficiently long winded that it doesn't make any difference and there is always a first segment ....
My manuals say it starts from the end of the TODR to gear up
correct
but I was always taught it's lift off (so basically the end of TORR) to gear up.
Suggest you change instructors
Is there a difference?
Certainly is - the flare distance to screen.
TOR is to the point where the wheels leave the ground
Common misconception.
TORA is declared by the airport folks and is just a number relating to runway construction details
Depending on the rules in play, TORR is either
(a) distance from start of takeoff to one-half the distance between lift off and screen, or
(b) distance from start of takeoff to one-third the distance between lift off and screen - probably no longer in vogue anyway, I'd guess
ie the aim is to give you a high probability of getting off the ground prior to the end of the higher bearing strength bits as defined.
Breaking the sometimes confusing requirements down into four segments
Minor point but, occasionally, you will find aircraft which use 5 segments