Please see section 1.2.2 on Space of Test Functions. On this page, it says
... $C^{\infty}(\Omega)$ is the set of functions on $\Omega$ with continuous derivatives of all orders.
I am not sure if "all orders" includes the zero order; i.e. what I am not sure if it says
$$u\in C^{\infty}(\Omega)\stackrel{def.}\iff u\in C^{k}(\Omega) \textrm{ for all }k\geq 0$$ or $$u\in C^{\infty}(\Omega)\stackrel{def.}\iff u\in C^{k}(\Omega) \textrm{ for all }k\geq 1.$$
I feel like the first one makes sense. But, in some literature, it's usually stated on the last one. For example at 1.3. DEF on page 6 [here $\mathbb{N}=\{1,2,\dots\}$ as indicated on page 4].
Short Answer
It doesn't make the slightest bit of difference. The two definitions give precisely the same space of functions.
Long Answer
The definitions "$C^\infty(\Omega)$ is the space of functions $u : \Omega \to \mathbb{R}$ (or $\to \mathbb{C}$) which have derivatives of all orders" and $$ u \in C^\infty(\Omega) \overset{\text{ def}}{\iff} u \in C^k(\Omega) \text{ for all } k \ge 0 \text{ (or $1$)} $$ are both a little imprecise or informal, though perfectly understandable and correct to all but the most pedantic of readers. That begin said, a more standard approach might be to first define $C^k(\Omega)$ as the set of functions which are $k$-times continuously differentiable, then phrase the second of the above definitions in terms of an intersection of spaces. More precisely, define $$ C^k(\Omega) := \left\{ u : \Omega \to \mathbb{C} \ \middle|\ \forall|\alpha|=k,\partial^\alpha u \in C^0(\Omega) \right\},$$ where $\alpha$ is a multiindex, and $C^0(\Omega)$ denotes the space of continuous functions on $\Omega$ (or, if you don't like that definition, pick your favorite definition for $C^k(\Omega)$). The phrase "$C^\infty(\Omega)$ is the space of functions which have derivatives of all orders" is then stated a little more precisely as $$ C^\infty(\Omega) := \bigcap_{k=1}^{\infty} C^k(\Omega). $$ Notice that this is an intersection taken over a decreasing sequences of spaces, as $C^k(\Omega) \supseteq C^{k+1}(\Omega)$ for all $k$. Thus the initial part of the intersection is irrelevant—only the terms in the tail contribute anything to this intersection. Hence it would be just as precise to define $$C^\infty(\Omega) := \bigcap_{k=0}^{\infty} C^k(\Omega) \qquad\text{or}\qquad C^\infty(\Omega) := \bigcap_{k=1}^{\infty} C^k(\Omega) \qquad\text{or}\qquad C^\infty(\Omega) := \bigcap_{k=10^{47}}^{\infty} C^k(\Omega). $$ Indeed, if $A$ is any finite subset of $\mathbb{N}$ (or of $\mathbb{N}_0$, if you prefer to include zero), then $$ C^\infty(\Omega) = \bigcap_{k \in \mathbb{N}_0 \setminus A} C^{k}(\Omega). $$ However, one would almost certainly not choose to define $C^{\infty}(\Omega)$ in this manner, as it adds superfluous sets and notation which reduce clarity and hinder effective communication. Similarly, choosing an initial index of $10^{47}$ would be quite idiosyncratic, and would leave the reader asking why the author chose such a peculiar starting point. On the other hand, it is generally hoped that the choice of an initial index of $1$ rather than $0$ (or $0$ rather than $1$) should cause readers no great difficulty.