There are different color models. They are 3D representations of the coordinates of different colors but using different axis and values.
RGB uses 3 linear axis. But some other uses 2 linear axis and one angular. The angular one is HUE.
There are different models and interpretations of how to grade the linear, for example, lightness vs. brightness.
Here are the channels (HSL) of a simple chromatic circle.
And here of a photo. The channel that is less abstract is the luminosity one.
According to the first description you posted, the Luminosity blending mode is using this channel to replace the values of the layer below.
Now some tests. I added a tiny diagram to the left of each so you can see which is over which and if it has Luminosity (L) or Color (C) blending modes.
The text says that the Luminosity Blending mode is the opposite of the color mode.
(A) The one on the left has the girl as Luminosity. You can see that the photo is only grayscale because is only a channel and is on top of the color wheel.
(B) The one on the right has the color wheel on top of the girl with color blending mode, and you can see that the girl's photo (hat) is in normal color (not grayscale)
The result is the same with the layers inverted. Girl-Wheel, Wheel-Girl.
There are differences in how software interprets the channels, even if they are called the same, so the result may vary.
Take a look at another post, where I try to compare some blending modes with real-life color manipulations.
Edition based on the comment.
(C) The circle on the left is the one I used, and you can see that the "darkness" (lightness) is overridden by the luminosity of the photo. The dark center is no more dark.
(D) On the right circle, the color of the circle is affecting, not the darkness (anti luminosity). You can even see that the circle is not dark at all. It is not showing its luminosity anymore.
will appear as grayscale only when it is on top of the other layer
The "other layer" is a circle with a white background, so it is gray on the zones with no "color" meaning a white background.
If it is on the bottom of another layer... you will not see it. It is behind... but something should be behind, probably your checkerboard background or something. There is always something behind it.
A blending mode is always the layer with the mode over another layer. This other layer can have or not another blending mode, but that is a separate thing.
Here is a screen capture of the program I'm using. I sent the layers that have the effect to the back. The photo on the left and the circle on the right. (F) You can see that now the circle is affecting the background gray and the white paper. The luminosity.
On the tiny corner of the left photo (E) , you can see it looks normal because it is only a luminosity, over a white and gray luminosity paper. So the photo is overriding that.