PA
it can only said that at Vy you want the highest forward component of velocity along with a vertical component that is based on the amount of excess THP available for climb,.
For max ROC (at VY) we must have maximum excess THP. We will use some of the available THP (the power required) to carry out the work of pushing the aircraft forward against the drag. Any remaining THP available can be used to push the aircraft vertically upwards. The ROC will be this excess THP divided by the weight. We do not necessarily require a high speed. If we had a machine of some kind in which excess power was maximum at zero TAS then that is the speed at which ROC would be maximum(provided we had some means of using that excess power to lift the machine vertically upwards).
and the greatest excess of THP and that would occur at Vimd in which you have the lowest drag there for the lowest thrust required at the highest TAS possible giving the greatest difference between THPr and THPa,...
This will be true only if the THP available is constant. This is not the case in any aircraft that I am aware of. With any type of prop the thrust decreases with increasing TAS. So the THP curve starts at zero (at zero TAS), increases with increasing TAS up to the TAS at which the rate of decrease of thrust outweighs the rate of increase of TAS. Above this speed the THP decreases. So having minimum drag will not necesarily give you maximum excess THP available.
Maximum excess power (and VY) will occur at VMD only if the thrust is constant at all speeds up to VMD, then decreases markedly beyond VMD.
If thrust remained constant above VMD then we would have the situation used to describe the theoretical behaviour of a Pure Jet in which VY is greater than VMD.
I think the question meant to say Vy not Vx as stated
The question being discussed did not ask for VX. It asked for the bast rate of climb speed (VY).
The text that you have quoted apears to be more concerned with how VX and XY vary with changes in altitude.
Because at that point in a piston [a constant power variable thrust machine--
A piston engine may be a constant
BHP machine, but a piston/prop is not a constant
THP machine. And in determining best ROC it is the excess
THP that matters.