Local Histogram Equalization (LHE) is a different algorithm. Histogram Equalization refers to a change to the image were you are aiming to achieve a "flat histogram", i.e., you redistribute the pixel values along the whole dynamic range yielding an image with the same number of pixels in the shadows, midtones and highlights. I wrote a module that contains this process. Also, in the new AdaptiveStretch process, with certain parameters you may achieve almost the same result. For normal images, due to the nature of the contents, Histogram Equalization usually gives good results (that may be improved by curves). In astronomical images, due the fact that background sky or objects dominate, this yields very poor results, so it should be avoided. The only reason to use HE is to aggressively stretch the background, and inspect the image (useful for placing DBE samples, or as an alternative to STF). The Local Histogram Equalization follows the same reasoning, but it tries to achieve flat histograms in small local areas. For example, take small 10x10px samples, and perform the HE there. At the end, you get an image with much greater local contrast (but the intensities loose correlation at large scales). There is a clamped version of the Histogram Equalization that prevents too aggressive changes to be made to individual pixels, and it is known as the CLAHE algorithm.
Histogram adjustments, on the other hand, is just made by selecting the black and white points of the image, and then using a midtones transfer function. In PixInsight, tha later is done with a rational interpolation algorithm, while in PS and other software the gamma function is used. There is also a third alternative, less common, that is to use a logarithm function. I also wrote a module that includes this as a manual process (along with the gamma function too). Also you may find it on the AutoHistogram process, as a selectable function. Usually I compare the results of stretching to the same values with all three functions. In most cases, I apply the result of the logarithm function, since it produces smaller stars, and a bit less global contrast than the rational interpolation.
