Nice tip about the battery. The video stopped just short of explaining WHY the modification makes it voltage independent. The answer is in the potentiometer vs. the rheostat used in the first drawing. The position of the setting will not react to the voltage because the overall resistance r1+r2 stays the same when the setting is changed. This is used in controls quite a bit. Even now when most of industry is handled digitally and remotely, there are still many systems with a good old "speed pot" on the control panel. I could go into much more detail but it would be even more off topic.
Cheers.
p.s. Pretty sure the rheostat at R3 is for calibration.