According to one article in Wikipedia, the odd convention of asterisks in L*a*b* is to distinguish between the current and an earlier version of the model, called Hunter's Lab. Since Hunter's is older, I assume Adobe is using L*a*b* and just simplifying notation.
I found another Wikipedia article, https://en.wikipedia.org/wiki/HSL_and_HSV, to be clearer than some other websites in clarifying the differences in terminology. Here's the relevant part:
This is consistent with a few other sources I found and suggests to me that when we talk about tonality adjustments affecting "saturation", what we really mean is "colorfulness". Luminosity-blend adjustments do affect saturation by changing the denominator, but they don't affect "colorfulness". Ditto, adjustments to the L channel. Normal-blend adjustments in RGB mode do affect colorfulness, as the photos above clearly show.Colorfulness: The "attribute of a visual sensation according to which the perceived color of an area appears to be more or less chromatic".[16]
Chroma: The "colorfulness relative to the brightness of a similarly illuminated white".[16]
Saturation: The "colorfulness of a stimulus relative to its own brightness".[16] [This explains why darkening with a luminosity blend or y using the L channel increases saturation]
Brightness and colorfulness are absolute measures, which usually describe the spectral distribution of light entering the eye, while lightness and chroma are measured relative to some white point, and are thus often used for descriptions of surface colors, remaining roughly constant even as brightness and colorfulness change with different illumination. Saturation can be defined as either the ratio of colorfulness to brightness, or that of chroma to lightness.