I'm with the same doubt that the author of this OP and I was thinking that the boundary of a class $C^k$ is locally a graph of a $C^k$ function, i.e., if $x_0 \in \partial U$, then exist $r > 0$ and a $C^k$ function $\gamma: \mathbb{R}^{n-1} \longrightarrow \mathbb{R}$ such that $\gamma(\mathbb{R}^{n-1}) = \partial U \cap B(x_0,r)$ for each $x_0 \in \partial U$. I would like to know if my thought about the definition it is equivalent to the definition given in the OP and, if it is, how can I prove this equivalence and why it's more interesting work with the definition of the OP that my understanding of $C^k$ boundary?
Thanks in advance!