The Cessna 172 POH has a table headed Landing Distance - Short Field. The speed used to attain the distances quoted is in fact the "normal" approach speed based on certification tests - ie 1.3 VS. The Boeing 737 "normal" approach speed for normal landings is the equivalent in principle to the Cessna figures.
In no case does the pilot deliberately reduce the approach speed below the stated "normal" approach speed as stated in the flight manuals (POH if you like). Therefore in reality the so called short field landing is always a normal landing speed and technique in order to attain the book landing distance figures.
During the war real short field landings required a certain speed reduction below the certification speed - usually 10% or even 20% lower. Landing on an aircraft carrier is an example of this where the approach is made at 1.1Vs instead of the normal 1.3Vs. (those figures are not necessarily correct but you get the idea, I hope). In those cases, it was slipstream provided by the propeller that kept the inboard wing area from stalling and if you chopped the power before the flare you would certainly fall out of the sky. Thus the true "short field" landing certainly required a powered approach "hanging on the prop" and of course as soon as the aircraft was flared the float would be minimal and the subsequent landing roll short. Just the sort of technique needed to land an aircraft into a field of unknown length but assuming the worst - very very short. Military trainee pilots were taught this technique until proved competent. These were not normal 1.3Vs landings that we know nowadays.
It wasn't until Flight Manuals were produced for civilian aircraft that certification requirements demanded a min speed of 1.3Vs for calculation of landing distances. Again there were probable exceptions to the method of stall speed calculation for purposes of landing speeds "over the fence".
All this boils down to there is no such thing as a short field landing where an approach speed significantly below the certification speed in the POH is routinely used. Thus if the POH speed is used correctly, then there will always be a float period to dissipate the speed (light aircraft such as Cessna types) before touch-down. That is because it is not the done thing to "plant" an aircraft on all three wheels at flying speed. There are obvious dangers to this - nose-wheel damage - bounce - and application of non-anti-skid brakes at relatively high ground speed.
If your instructor demands an approach at less than the POH recommended approach speed (1.3Vs) then it is both illegal and reckless. It is never done in a airliner or business jet, so why should it be done in a trainer?