Derivative Notation as a factor
As stated in the text, it's called operational notation, i.e. $D$ is an operator being applied to a continuous-time signal:

$$D^mx(t)=\frac{d^mx(t)}{dt^m}$$

In discrete time you have the operator $T$, defined by

$$T^mx[n]=x[n+m]$$

(see e.g. here).

What is often done by people is that they use the complex Laplace transform variable $s$ instead of $D$, or the $\mathcal{Z}$-transform variable $z$ instead of $T$. This is what Robert Bristow-Johnson referred to in his comment as 'sloppy abuse of notation'. I would even go further and would call it simply wrong. The use of different letters for $D$ and $T$ may seem an unimportant triviality, but the real problem here is that you mix different domains (Laplace/$\mathcal{Z}$ transform domain and time domain), which can be a great source of misunderstandings and errors.

I personally do not see a great advantage in the use of this type of operational notation for engineering applications. It rather makes things more confusing for beginners because they have a tendency to mix domains anyway, and operational notation seems to suggest that this is legitimate.