I have not read the whole thread but the answer seems to be simple: it is the saturation slider in the contrast brightness saturation module. The module works in Lab.
However, I also use both programs and from my experience I can tell you that although in theory the two sliders are the same thing they just do not produce the exact same result. They somehow calculate differently. I personally like the way the saturation slider indt pushes the colors better. But white balace also has an influence on this. It is complicated…