I understand that the add is adding that color, and multiply is coloring the brightness. But I don't understand why the lighten only colors dark areas. Unlike the name, why does the lighten add color to the dark part?
Lighten = pick the highest value color
As shown in the manual:
Blender is using the transformations as described in the Gimp specifications:
Lighten only mode compares each component of each pixel in the upper layer with the corresponding one in the lower layer and uses the larger value in the resulting image. Completely black layers have no effect on the final image and completely white layers result in a white image.
So it's basically using the highest value color, thet's why it's more probable to affect the darker areas as your shadows