Why is the relation of height and temperature different when calculating altitude? I'm trying to grasp the essence of altitude calculations. One thing I can't figure out in my head is this:

In the International Standard Atmosphere (ISA) the temperature of the air decreases with the altitude, which is called the lapse rate. This is an approximate of 1.98 C (degrees Celsius) or an even more approximate 2 C per 1000 feet. This is usually described through dry adiabatic lapse which means simply that when you go up the pressure rises and the air expands thus through the ideal gas equation it makes sense for the temperature to drop as well.

However when calculating the density altitude we need to "correct" the pressure altitude with this formula:

DA = PA + (118.8 ft/C) x (OAT - ISA temperature)


(from https://en.wikipedia.org/wiki/Density_altitude#Calculation)
Here the height change per degree Celsius is only 118.8 feet (approx 120 ft).

My question is this: Why is the relation of (change rate) height and temperature different? With just "going up" it changes 2 C / 1000 feet (ie 500 ft/C) but when adjusting for temperature it changes 120 ft/C).

These both are simple linear equations but have different coefficients.
The equation is saying, that for every degree difference from standard temperature, the difference between pressure and density altitude grows by 118.8 ft. A completely different factor from the lapse rate.