Here is an 8 minute video that demonstrates the luminosity painting technique. In the video, the goal is to increase local contrast by lightening (i.e. dodging) certain lights and darkening (i.e. burning) certain darks.
The dog in the OP is a good subject for this technique. The streak of reflected light on the dog's coat was already there. You can enhance it by dodging through a luminosity selection which selects the streak. To exaggerate the effect, you could then select for the surrounding dark fur and burn through that.
An example that would not be appropriate for luminosity painting would be a portrait with the face evenly illuminated by flash. Say you wanted to create the effect of split lighting (one side darker than the other). Here, luminosity painting would not help you because there is no mask that would be selective for one side or the other. However if the original portrait had been taken under split lighting, you could use luminosity painting to exaggerate (or lessen) the effect.