What is meant by "Cubic Difference Frequency"?
After skimming through the paper, I can see more clearly now. The measure $D_3$ quantifies the relative strength of the 3rd order intermodulation product. If two sinusoidal signals with frequencies $f_1$ and $f_2$ are input to a non-linear device (such as a microwave amplifier, as referred to in the paper), there will be intermodulation products at the output. Third-order intermodulation products occur at frequencies $2f_1+f_2$, $2f_1-f_2$, $2f_2-f_1$, and $2f_2+f_1$. Problematic are the ones at frequencies $2f_1-f_2$ and $2f_2-f_1$ because they can end up in the frequency band of interest.

The quantity $D_3$ is defined as the ratio of the power of one of these undesired frequency components and the power of one of the desired frequencies (in dB) at the output of the nonlinear device. There are four possible definitions:

$$D_{3,1}=-10\log\left(\frac{P_2(2f_1-f_2)}{P_2(f_1)}\right)\\
D_{3,2}=-10\log\left(\frac{P_2(2f_1-f_2)}{P_2(f_2)}\right)\\
D_{3,3}=-10\log\left(\frac{P_2(2f_2-f_1)}{P_2(f_1)}\right)\\
D_{3,4}=-10\log\left(\frac{P_2(2f_2-f_1)}{P_2(f_2)}\right)$$

where $P_2(f_x)$ is the output power at frequency $f_x$. These four definitions do not necessarily result in the same value for $D_3$. It's just important to specify which of them is used, and to consistently stick to that definition.

To answer your questions: no, $D_3$ is not the power of the 3rd order intermodulation product, but it is the ratio (in dB) of that power to the power at one of the signal frequencies. This quantity is defined based on the assumption that the input signal is the sum of two sinusoidal signals, but it is used to quantify the strength of 3rd order nonlinearities at the output of a nonlinear system regardless of the input signal.

Concerning the power compression factor, you can view the changes $\Delta P_1$ and $\Delta P_2$ as infinitesimal changes, and so the question if $P_1$ and $P_2$ are the values before or after the change becomes irrelevant. At a given frequency, an ideal linear amplifier would satisfy

$$P_2=g\cdot P_1$$

where $g$ is the gain factor at the given frequency. So you have

$$g=\frac{P_2}{P_1}=\frac{dP_2}{dP_1}\Longrightarrow\frac{dP_2}{dP_1}\frac{P_1}{P_2}=1$$

That's why the power compression factor

$$c=1-\frac{dP_2}{dP_1}\frac{P_1}{P_2}$$

is a measure for the nonlinearity of power amplification.