If $\partial\Omega$ is of class $C^1$, does it need to be the boundary of a $C^1$-regular domain? Let $d\in\mathbb N$ and $\Omega\subseteq\mathbb R^d$.
In Theoretical Numerical Analysis, the notion of $\partial\Omega$ being "of class $C^1$" is defined in the following way:

On the other hand, in Introduction to Smooth Manifolds (p. 120) $\Omega$ is said to be "$C^1$-regular" if $\Omega$ is a $d$-dimensional properly embedded $C^1$-submanifold of $\mathbb R^d$ with boundary. In that case, the manifold and topological boundary coincide and $\partial M$ is a $(d-1)$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$.

So, I wonder how these two definitions are related. In the former, $\Omega$ is assumed to be $\mathbb R^d$-open, while in the latter, $\Omega$ being properly embedded is equivalent to $\Omega$ being $\mathbb R^d$-closed. Can we generally show that if $\partial\Omega$ is "of class $C^1$", then $\partial\Omega$ is a $(d-1)$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$? Can we even show that $\partial\Omega$ must be the boundary of a "$C^1$-regular domain"?

EDIT:
Clearly, if $\Omega$ is open, then it is a $d$-dimensional embedded $C^\infty$-submanifold of $\mathbb R^d$. So, I've got the feeling that together with assuming that its boundary is of class $C^1$ yields that $\overline\Omega$ is a $d$-dimensional properly embedded $C^1$-submanifold of $\mathbb R^d$. Can we in fact show this?
 A: First of all, you quoted the definition in my book ISM incorrectly. I don't mention "$C^1$-regular domains" at all; instead, I only treat smooth (i.e., $C^\infty$) regular domains, which I define as properly embedded codimension-$0$ [smooth] submanifolds with boundary (not submanifolds as in your version).  And I don't understand what the relationship between $M$ and $\Omega$ is supposed to be in your version of the definition.
My definition of regular domains can easily be extended to domains of class $C^k$: a $\boldsymbol{C^k}$-regular domain in $M$ is a closed subset $D\subseteq M$ that is a codimension-$0$ topological manifold with boundary in the subspace topology, endowed with a $C^k$-structure such that the inclusion map is a $C^k$ embedding.
If $\Omega\subseteq \mathbb R^d$ is a $C^1$-regular domain by the definition 7.2.1 that you quoted, then $\overline\Omega$ is a $C^1$-regular domain by the definition I just gave. Conversely, if $D\subseteq \mathbb R^d$ is a bounded regular domain by my definition, then its interior is a $C^1$-regular domain by your definition. (It's a matter of taste whether one considers a "domain" to be an open subset or a closed subset.) I don't have time to write out a complete proof, but the basic ingredients are $C^k$ versions of the immersion theorem for manifolds with boundary (Theorem 4.15) and the implicit function theorem (Theorem C.40), both of which follow from a $C^k$ version of the inverse function theorem (Theorem C.34). The proof of the inverse function theorem that I give in my book can easily be adapted to the $C^k$ case, just by stopping the induction when you have proved that the inverse map is of class $C^k$.
