Dynamical system
What is a second-order dynamical system?
Let's firstly clear up some confusion about the term dynamical system.
The term in your document refers to the mathematical definition, where a function describes the time dependency of a point (or multiple points) in space.
A first-order dynamical system involves one derivative. It can be described by one characteristic. An example of such a system is the current-voltage relation of a capacitor in electronics.
A second-order dynamical system involves two derivatives. It can be described by two characteristics. An example is a mass-spring-damper system, which is a mechanical system.
In mechanics, we refer to dynamic (not dynamical) systems more specifically when we describe time-dependent movement of mass in space. According to Newton's law, we can instantaneously apply finite acceleration by applying finite force. We are eventually interested in position, which is two derivatives apart from acceleration. Every mechanical dynamic system is thus at least a second-order dynamical system. In fact, a mechanical system with $n$ degrees of freedom is a $2n$th-order dymamical system: the state of every degree of freedom can be described by a position and velocity, as both variables cannot be changed instantaneously.
It is possible to express a $2n$th-order dymamical system as a set of $n$ second-order dynamical systems or a set of $2n$ first-order dynamical systems, but it is incorrect to say that a (mechanical) dynamic system with multiple degrees of freedom is a second-order dynamical system. The "robot and environment" in your example however can be correct if they refer only to the robot-environment interface, described by a single position and velocity variable.
Control
For purposes of control, it is thus useful to have both position and velocity as feedback because neither of them can be changed instantaneously: they essentially define the (instantaneous) state of the dynamic system, i.e. they are used as state variables.
Example - If you're the driver of a car and $100~\mathrm{m}$ from your target, then what are you (the controller) going to do: accelerate or decelerate? This question is difficult to answer unless you know your current velocity. If your velocity is zero, then obviously you want to accelerate or you're not getting anywhere. If your velocity is $200~\mathrm{km/h}$, then, well... good luck finding that brake pedal.
All nice and dandy, but you might have found in literature that some mechanical systems are controlled by using only position variables. This is reasonable if you can assume that velocity can be changed instantaneously, which is a valid assumption if systems are to be operating excruciatingly slow, where maximum velocity is simply capped or if the system suffers from enormous amounts of viscous friction (where force is essentially proportional to velocity instead of acceleration).
Example - If you're the driver of a car and $100~\mathrm{m}$ from your target, and if you know that the maximum velocity of the car is $5~\mathrm{km/h}$, then what are you (the controller) going to do: accelerate or decelerate? You will obviously accelerate, if not already driving at the excruciatingly slow $5~\mathrm{km/h}$...
Trivia: In the realm of mechanics, for systems that are modelled to not or partially include velocity related terms, the term pseudo-dynamic system has been coined.