I think that, in general, images that are in grayscale should be left as they are, and colored images should be adjusted for a neutral background color such as white or black. As one can see from your examples, adjusting the background requires adjusting the colors of the rest of the image. The first image has a cyan tone to it, and removing that tone reduces the blue in all of the colors throughout the image. I believe this looks better than the original, but I suppose there will be circumstances where it doesn't. As long as the adjustment looks good overall, I think it should be used, 99% of the time.