There's no such thing as the differential operator, but there is a class of things that are, collectively, called differential operators.
Imagine that we're working in a function space such as $\mathcal C^\infty(\mathbb R)$. Then $\mathcal C^\infty$ is a vector space (by pointwise addition and multiplication by constants), and a linear map from $\mathcal C^\infty$ to itself is called an operator. The space of operators is itself a vector space, and we can make it into a non-commutative ring by declaring "multiplication" of operators to be functional composition (since the composition of two linear maps is itself a linear map).
We can decide on a subring of "simple" operators, such as all operators that multiply by a constant, or all operators that work by pointwise multiplication by a fixed function. Which operators are "simple" depends on the application. (Beware that this sense of "simple" is not standard; I'm defining it for the occasion here).
Now, if our function space is sufficiently well behaved, the differentiation operator $\partial$ (which takes a function to its derivative) is a linear map from the function space to itself, and therefore an operator. A differential operator is then an element of the least subring that contains the simple operators and the differentiation operator. Such an operator is always the value of some polynomial expression with $\partial$ as the variable and simple operators as coefficients, and it is in this sense that your reference considers a differential operator to be a "function of" $\partial$.
In particular, the differentiation operator is itself a differential operator.
Often one would not consider the simple operators themselves to be "differential operator"; but in other cases one might want to consider them trivial cases of differential operators. That, again, depends on the application, and on authorial preference.
In more complex functions spaces, such as $\mathcal C^\infty(\mathbb R^n,\mathbb R)$ one might want to consider several basic differentiation operators, such as $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial y}$, and a differential operator is then something that can be written as an expresion involving those. Things get yet more involved on smooth manifolds, but the basic principles stay the same.