So my best guess for the results of the youtube experiments is that the shade being used to create an area without moonlight is affecting the amount of radiative cooling at that area, resulting in it being less cold.
If that's what's happening, I guess the key then is to ensure that the shade is at a sufficient distance from the areas being measured - making use of a mountain, for example.
In any case, here's a couple of videos (from the same poster) (and the last of these videos I'll be posting in this thread) that again show the same results.
The first one begins at 2:35
, showing a metal motorcycle ramp that has been left for around 3 hours:
It shows, at around 4:00
, that the area that has been in the moonlight has a temperature of around 53.6F, while, at around 4:15
, the other end of the ramp has a temperature of around 56.1F.
There is a shade and a light that he's using, which can be seen at 2:37
. The light is slightly closer to the moonlit side, while the shade is slightly closer to the non-moonlit area:
Could this proximity of the shade to the left-hand side of the motorcycle ramp account for the ~2.5F change in temperature?
In his second video he places three objects on the ramp: a leather jacket, a metal ruler, and a piece of wood. He begins to take his readings at 3:30
His results are as follows:
Leather jacket: about 1 degree colder in the moonlight (49 to 50F)
Piece of wood: ditto (51.5 to 52.5F)
Metal ruler: about 4 degrees colder (52.5F to 56.5F)
Again, he is using shades to block off the moonlight, as shown at 1:15
Does the proximity of these shades account for the differences in temperature? I suppose one way to test for that would be to use the same set-up on a no-moon night and see if the differences were still the same.