Let's start with the simplest fact. With nothing else going on, air in our atmosphere will always be cooled at 5.4°F/1000ft (or the more widely used 9.8°C/km), due to it expanding in size (often times we'll throw a negative sign: -5.4°F/1000ft... this is just to emphasize that these lapse rates are about the air COOLING). Every bit of the rising air cools at that rate (the oxygen, the nitrogen, even any water vapor in the air). All of it will cool at 5.4°F/1000ft. All rising air always expands at this same rate.
However sometimes there is a second complicating factor: the rising air may cool to the point where it is holding its maximum quantity of water vapor (because when colder, air cannot hold as much water vapor).
When it reaches this point, it is saturated, and any more cooling/rising will force some of its water vapor to condense instead into liquid droplets. For this gaseous water vapor to become a liquid, its particles must expel quite a lot of energy. And that energy gets released into the air.
So once air hits this saturation point, though its expansion continues to have a -5.4°F/1000ft impact on the temperature, it also has a second factor, that released phase change energy, warming the air some. The result is that saturated air doesn't cool as fast.
- Whenever air isn't saturated, it cools at the 5.4°F/1000ft (9.8°C/km). We call this the dry adiabatic lapse rate because nothing else is altering the process. We aren't assuming 0% moisture. It doesn't matter, as long as it's not 100% moisture. Water vapor cools just like every other molecule unless saturated. The first 6 pages of these notes go into intimate detail for why the value is 9.8°C/km, but basically it depends only on the values for gravity and the specific heat capacity of water, plus any error in the hydrostatic approximation... all of which are minuscule for large air masses in the troposphere. For all intents and purposes, the dry (unsaturated) adiabatic lapse rate on Earth depends on nothing, it is always 5.4°F/1000ft (9.8°C/km)
- Then when air is saturated, because the condensation process hinders the cooldown by adding its energy, we call it the moist adiabatic lapse rate to distinguish it. Picture moist as having an image of water DROPS, which you only get when saturated, due to condensation. Moist and dry both have some water vapor. But only moist has water droplets. The moist adiabatic lapse rate has a smaller value because it is cooling LESS overall (due to the warming impact from the latent heat release). But it is still cooling, just less. The moist adiabatic lapse rate varies depending on how much water is actually released (which is based upon the amount of water held, or consequently, the temperature). If you really wish to dig into the fine details, this AMS link gives the formula. But generally the moist adiabatic lapse rate is around 5-7°C/km, or 2.5-4.0°F/1000ft, in the low-levels of the atmosphere.