In his CNX2 book, Jason Odell states that Levels and curves "affect color," while the LCH editor (operating on "Master lightness") affects only luminance, "preserving color." Since reading his book, I have used LCH exclusively for modifying both global and local contrast. I have not fully understanding why, but I trust that Jason knows what he's talking about.
Of course "color" is a rather imprecise term, so here are my findings, based on experiments with the LCH color model.
* When you push the black point in Levels and curves, Chroma increases. That is, colors become more saturated; they do not tend toward black. * When you push the black point in the LCH editor, Luminence decreases, and colors go toward true black.
If you push the white point in curves, and chroma decreases. Colors become less saturated, tending to white. If you push it in LCH, luminence increases, and colors also go to white.
Of course I wonder whether this it makes any practical difference which tool you use when working on a real photograph like the OP's, or only when doing the most demanding studio work.