Brightness is luminance expressed in perceptual units. Luminance is Y from CIE XYZ, brightness is Jz from JzAzBz or L from CIE Lab or CIE Luv.
Basically, it all falls back to:
- some metric of value : (linear) luminance or (perceptual) brightness,
- some metric of the distance between current chromaticity and the achromatic color having the same value : the chroma, which can be either linear or perceptual.
Luminance/chroma and brightness/chroma are both a set of perpendicular coordinates, only scaled differently.
Now, from brightness/chroma to saturation/purity (or actually, I learned recently that purity is actually called brilliance since the 1950’s, might change the name), we just rotate things :
(from The Difference Between Chroma and Saturation | Munsell Color System; Color Matching from Munsell Color Company)
The lines of equal saturation are the oblique ones, so the saturation setting opens or closes their angle like a flower, while purity moves along those oblique lines.
Changing luminance/chroma to saturation/brilliance changes an important thing: if you “de-chroma” red, you move to grey at same luminance, if you desaturate red, you move to pink. Any kindergarten kid knows that adding white in red paint gives pink, yet it’s an impossible color to get in photography, because any saturation algo actually uses chroma.