I am trying to draw some relations between two (apparently) different definitions. The first one is from Evan's book:
Let $U \subset \mathbb{R}^n$ be open and bounded. We say the boundary $\partial U$ is $C^k$ if for each point $x_0 \in \partial U$ there exists $r>0$ and a $C^k$ function $\gamma: \mathbb{R}^{n-1}\mapsto \mathbb{R}$ such that - upon relabeling and reorienting the axes if necessary - we have: $$U\cap B(x_0, r)= \{x \in B(x_0,r) | x_n > \gamma(x_1, \dots, x_{n-1})\} $$
The second definition would be the defintion of a regular domain: More specifically, a properly embedded submanifold with boundary and codimension zero. My guess that these two definitions might be connected comes from a proposition in Lee's book which states that for a smooth function $f: C^{\infty}(\mathbb{R}^n)$ every sublevel set corresponding to a regular value $b$ of $f$, i.e., every set of the form $f^{-1}((-\infty, b])$ is a regular domain in $\mathbb{R}^n$.
Overall I have two questions:
(1) Is the Evan's definition equivalent (apart from the restriction to k-times instead of $\infty$-times differentiability) to the statement that $\overline{U}$ is a regular domain in $\mathbb{R}^n$?
(2) Why does the Evan's definition use closed balls instead of open balls, i.e., would the meaning of the definition change if we replaced $B(x_0,r)$ with $B^{O}(x_0,r)$?