According to Wikipedia and some other sources, a hypersurface is a manifold of dimension $n-1$ embedded in an ambient space of dimension $n$ which is simple to understand.
Now here https://homepages.warwick.ac.uk/staff/C.M.Elliott/DziEll13a.pdf the Definition 2.1 has a different definition of a hypersurface:
Let $k \in \mathbb N \cup\{\infty\}$. $\Gamma \subset \mathbb R^{n+1}$ is called a hypersurface if for each point $x_0 \in \Gamma$ there exists an open set $U \subset \mathbb R^{n+1}$ containing $x_0$ and a function $\phi \in C^k(\Gamma)$ with $\nabla \phi \neq 0$ on $\Gamma \cap U$ and such that $$U \cap \Gamma=\{x \in U \,\vert\, \phi(x)=0 \}$$
Does this have a synonymous meaning as the definition from say Wikipedia? If so, how? I don't have any intuition what the latter definition should mean.