Judging from the little interactive diagrams in your link #1, the author has set up the CIELAB and RGB color spaces with different white points, without doing any kind of chromatic adaptation transform in between. As a result, the widget presents a sort of pale orangish-brownish color as 'neutral' and all the slices related to hue and chroma are noticeably off. It’s an unfortunate oversight, because it makes them much more confusing than they would otherwise be.

In general, I’d recommend people use the CIECAM02 color space for most of the purposes being discussed in this thread. Here’s my implementation: https://github.com/jrus/chromatist