In Partial Differential Equations by Evans, the author defines a set as having a $C^k$ boundary as follows:
Definition. Let $U \subset \mathbb{R}^n$ be open and bounded. We say the boundary $\partial U$ is $C^k$ ($k \geq 2$) if for each point $x^0 \in \partial U$ there exists $r > 0$ and a $C^k$ function $\gamma: \mathbb{R}^{n-1} \to \mathbb{R}$ such that--upon relabeling and reorienting the coordinate axes if necessary--we have $$ U \cap B(x^{0},r) = \{x \in B(x^0,r) : x_n > \gamma(x_1,\ldots,x_{n-1}) \}. $$
I would have assumed that having a $C^k$ boundary would mean that the boundary has a $C^k$ parameterization, that is:
$\partial U$ is $C^k$ if there exists a set $D \subset \mathbb{R}^{n-1}$ and a $C^k$ map $\Phi: D \to \mathbb{R}^n$ such that $\partial U = \{\Phi(x): x \in D \}$.
Are the above definitions equivalent? I'm pretty sure they are equivalent for dimensions $n = 2,3$, but I'm not sure about arbitrary $n$.