I think you’re spot on target with your theory @uberbatman. I once knew a clown who could do the same thing with his asshole. It was even more pronounced than from mouth, because he could control more than just fart temperature, but also the sound, the pungent weight of the fart, the dilution of the stench, and even the cling time, depending upon the wind of course.

Dude really knew how to clear a room.

When you purse your lips an exhale a narrow, rapid airstream, that jet of air pulls in lots of the surrounding room air (this is either Bernoulli’s principle or the Venturi effect, I don’t remember). By the time it reaches your hand, it’s mostly room air, which is likely cooler than your hand. If you puff out a gust of breath through open lips, the surrounding air isn’t pulled along, so it’s mostly heated breath that hits your hand.

I think the amount of heat(lost) in exhalation breath will depend on how long it’s in your lungs and windpipes, not how you let the air out of your body.
@thorninmud has a good answer for why it may not seem that way.(Venturi gets my vote)

Breath fast or slow. The sooner you let it out, the cooler it will stay. And the longer you hold it in, the warmer. Blowing on your hand involves the wind chill factor. Your breath, and the other drier air that gets caught up in it, increases the evaporation of the moisture on your skin, which make it feel cooler. (One of the principals of basic air conditioning here: When water evaporates, it absorbs heat creating the cooling effect.)

The faster you breath out onto your hand, the more wind chill effect you get, the faster the moisture evaporates, the more cooling takes place. How high or low the humidity is of the air passing across your hand affects the wind chill, too. The main thing you’re dealing with here is the wind chill factor.