I won't attempt any definition, but a practical approach is to consider luminosity the perceived brightness based on visual spectral sensitivity. A fully saturated yellow is perceived as brighter than a fully saturated blue. The Lab color model takes this into consideration, by separating the luminosity and chroma components.
Brightness OTOH, at least as defined in Photoshop, is based strictly on RGB values. Equally saturated yellows and blues have equal Brightness.
The Luminosity blend mode / luminosity masks in Photoshop are based on the Lab L channel.
This illustrates the difference:
So if you want Luminosity, as opposed to Brightness, use the L channel in Lab, or Luminosity blend mode. Here's a more real life illustration:
(edited for clarity)
I am pretty sure the Luminosty in Photoshop is defined as: 0.3*R + 0.59*G + 0.11*B
A little experiment:
1) I created a new picture and put a color to a layer.
2) There are few ways to find the RGB values of the color (Color picker, eyedroppertool, info palet). It's easy to compute with hand the lumonisity.
3) Open Histogram palet. Luminosity value is given here by Photoshop if you choose the Luminosity channel.
My conclusion is: Above formula for Luminosity is what Photoshop uses.
I think I understand if a change in luminosity also effects saturation if Hue is constant.
If I simply think about these colorspaces: H, S, B and Hue, Saturation, Luminosity
And define Hue as constant... Then S and B equals to S and L (to keep the same color).
As Brightness and Luminosity are not the same, the saturation has to change.
Some sites imply Luminosity and Brightness are interchangeable (the way I read it at least).
This is meaningless. RGB values for any given color depend on the color space it's encoded in. You'd get different results in sRGB, Adobe RGB and ProPhoto, or any other imaginable color space.
For the same reason, HSB and HSL are both undefined in themselves. They are color models, not color spaces.
That said, absolute saturation naturally decreases as you move from the color sphere's "equator" towards one of the "poles". On either pole, it's obviously zero whatever the original hue.
Edit: just to be clear, Lab is a fully defined color space. HSB/HSL are color models, only defined in terms of what color space they refer to.