Definition of Surface-based atmospheric duct
Both situations are possible. Normally, the atmosphere gets colder and more rarefied with height, and cold air holds less moisture than warm air. Inversion layers trap the lower air mass closer to the ground, so there is a reversal of what you might normally expect: temperature and humidity rises as you get higher, rather than the normal decrease. Inversion can occur in several ways, for example, if a mass of warm air moves in over a cold surface, an inversion will take place. Bothe statements you quote are correct, but the second one is more usual.