Here is my justification for choosing the Y variable:
The way I think about it is not how to transform some god-given variable like cd/lm but how to create well-anchored a domain where equal spacing have easy to interpret physical meaning, which also conveniently compresses the range. This part has nothing to do with perception of brightness (if you don’t want it to).
With the Y chosen, √cd for X is the logical and Y-matching choice to compress it as well. While we can debate the perceptive interpretation of it, the screen you’re looking at already uses gamma transform, which is close to it, so it’s probably not too far off from reality and practice for photopic vision.
The red contour lines for the ‘reach at 1 lx’ also work nicely with these two choices - one can increase the reach by either pumping out more lumens or tightening the beam and this family of hyperbolas can tell you what’s easier. BTW while unlabeled, the contours start at 100 m and go to 2000 m, in 100 m steps.