Hard to tell if this is auto generated but I'd guess so. There's a somewhat egregious error. In the Helmholtz–Kohlrausch effect section the site states"
"The illusion where increased saturation (or chroma) of a color is perceived as an increase in the color's lightness."
But that is notably wrong. Lightness is a tonal metric (closer to white, more lightness) and is defined as such on that site as well.
In the H-K effect, emitted light of greater saturation (less lightness) is perceived as higher luminance (brighter) than emitted light of greater lightness.
A more accurate statement would be "The effect whereby colors of greater saturation than, but equal luminance to, a less saturated reference color are perceived to be of greater luminance that the reference color." (I'm sure that could be tightened up.)
There is some conflation here between an art definition and the colorimetric definition. In colorimetry:
Luminance is a spectrally weighted measure of physical light, it is not a perception, and luminance is linear to physical light.
Lightness is the perception of luminance, along with darkness and brightness. In colorimetry, the term lightness is used to distinguish the perception of light (which is nonlinear to physical light) as opposed to luminance which is a linear measure of physical light.
Luminance is denoted Y (relative) or L (absolute), Perceptual lightness is L* (Lstar) or J, brightness as Q (in most models).
As you noted, the term lightness can also be used as a tonal metric (i.e. in art). However in colorimetry, the closeness to white, or achromatic, is described as less purity (less chroma or less saturation).
And for the record, the site is not auto-generated. Nate Baldwin created that site. Nate is also the developer behind Adobe Leonardo, the color palette utility.
As for the Helmholtz–Kohlrausch phenomenon, here is the official definition from the CIE - International Commission on Illumination
Change in brightness of perceived colour, produced by increasing the purity of a colour stimulus while keeping its luminance constant within the range of photopic vision
Note 1 to entry: For related colours, a change in lightness can also occur when the purity is increased while keeping the luminance factor of the colour stimulus constant.
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The CIE points out that the change in perceived lightness occurs with increased saturation (purity).
"The illusion where increased saturation (or chroma) of a color is perceived as an increase in the color's lightness."
But that is notably wrong. Lightness is a tonal metric (closer to white, more lightness) and is defined as such on that site as well.
In the H-K effect, emitted light of greater saturation (less lightness) is perceived as higher luminance (brighter) than emitted light of greater lightness.
A more accurate statement would be "The effect whereby colors of greater saturation than, but equal luminance to, a less saturated reference color are perceived to be of greater luminance that the reference color." (I'm sure that could be tightened up.)
A very different thing.