If you place the blackbody radiation curves for Sun & Earth at its average temperature of about 15 °C on the same graph, you'll find the they meet around the wavelength of 4.3 microns - which is between the peaks of solar (.47 microns) of earth IR (10.7 microns) emission. The Fluke 62 measures IR radiation at 6.5-18 microns:
http://support.fluke.com/find-sales/download/asset/2437622_a_w.pdf
Thus solar radiation is probably insignificant - especially pointed away from the sun early or late in the day - obviously no problem at night.
So yes - the temperatures of low & mid level clouds are often about 10-15 °C - which is probably what you are measuring to a large extent. They emit IR radiation very similar to blackbodies at that temperature. It may even register a bit too high in that case because the emissivity of the clouds is greater than objects the instrument is intended to measure, and because of aerosols below. When you point it to the sky, you are most likely getting an integrated average of sorts of the temperature of particles in the atmosphere to space. The average temperature of the atmosphere over the globe is about -20 °C, but the particles are concentrated near ground. I would bet it does indeed register higher when the air aloft is warmer and lower when the air aloft is colder. Play around with it, look at the weather maps, and see. You need nights with absolutely no clouds and probably a similar amount of aerosols and humidity.
You seem to have a pretty good idea about what is causing the readings you are getting, and the numbers seem pretty good - may even be more accurate than pointing it at an object right in front of you - just kidding :) :) :)
Regarding the post below. Water vapor satillite images are from measurements at specific bands wihtin the range of about 6-8 microns and not a spectrum as you are measuring, and are a view down from space - which differs from what you are doing. You can easily test the water vapor only theory during a night with a variable low cloud cover - clear & cloudy periods would be the same. From what you already posted I seriously doubt that.