I agree there is nothing wrong with using the default amount of water, usually, but it can lead to odd results in some (less common) soaps (like significant warp in CP olive soaps due to excess water, which I discovered very early on, due to the recipes I was making).
If you get into the habit of using the lye concentration or water:lye ratio from your first soaps, later adjustments to the water amount are easy to calculate and the impact of changes to the lye concentration become predictable, even when different oils (recipes) are used.
For your extra hot method of HP (up to 213F measured at the surface, at one point in the video!), I would expect that you need to add some water to the recipe, to account for evaporative loss.
With the CP and SVHP comparison recipes that I've made, no change to the water in the recipe is necessary (near zero evaporation). I have used about 33% lye concentration (or roughly 2 parts water to 1 part NaOH) for both methods.
So it is very method dependent, with HP potentially (but not always) needing more water.
A way to know how much water has evaporated, is to weigh your soapmaking equipment (pot, tools) at the start and finish of soaping, and record the difference.
Do the same with your molds, weigh them at the start and after your soap is in the mold, and record the difference.
The amount of "waste batter" is the difference in your tool weights.
The "molded soap weight" is the difference in your mold weights.
The waste batter and molded soap weight added together is your total soap weight.
The weight difference between the original recipe amount and the total soap weight is the amount of water that has be lost to evaporation.
When you know how much you water you have lost to evaporation, you can convert that to an evaporation percentage (the percentage of water you expect to lose during the cook), and from this you can make your recipe in CP and be able to predict the necessary adjustment to the water amount.
This comes in particularly handy if you are taking a CP recipe and converting it to your HP process - you will know how much extra water to add.