I'm studying manifolds and I am feeling unconfortable with one definition in particular. The definition is given below:
.
Considering this definition, the candidate of manifold I'm using is the set
$$S = \{(x,|x|)\ |\ x\in (0,1)\}$$ with the Topology inherited from $\mathbb{R}^2,$ which is depicted below: 
This could be considered a smooth manifold because the inverse of $$f: \begin{align}(0,1)& \rightarrow S \\ x& \mapsto (\text{sign}(x)g(x),g(x)) \end{align},$$ where $$g(x) = \left\lbrace \begin{align} &e^{-\dfrac{1}{x^{2}}},& x \not= 0\\
&0,& x=0
\end{align}\right.$$ is a chart. Now, to make the atlas $\{f^{-1}\}$ maximal, we just need to enlarge the maximal atlas, as stated by the author.
What is wrong here? Am I missing something?
You are confusing yourself with two different concepts: that of a smooth manifold and a smooth submanifolds.
Consider the map $f: \mathbb R \to S$, $x\mapsto (x, |x|)$ and $g:S\to \mathbb R$, $(x, |x|) \mapsto x$. Both are continuous and are inverse to each other. Thus $S$ is homeomorphic to $\mathbb R$. Since $\mathbb R$ obviously has a smooth structure, the same is true for $S$.
On the other hand, $S$ is NOT a smooth submanifold of $\mathbb R^2$. If you don't have the definition of a smooth submanifold yet, then understand this as "there is no smooth structure on $S$ so that the inclusion $\iota :S \to \mathbb R^2$ is smooth and immersed".