Let $\mathcal C$ be a class of functions between Banach spaces, $d\in\mathbb N$ and $k\in\{1,\ldots,d\}$. We say that $M\subseteq\mathbb R^d$ is a $k$-dimensional embedded $\mathcal C$-submanifold of $\mathbb R^d$ if $M$ is locally $\mathcal C$-homeomorphic$^1$ to $\mathbb R^k$.
On the other hand, we say$^2$ that $\partial M$ is of class $\mathcal C$ if for each $x\in M$, there is an open neighborhood $\Omega$ of $x$ and a function $g:\mathbb R^{d-1}\to\mathbb R$ of class $\mathcal C$ with $$\Omega\cap M=\{x\in\Omega:x_d>g(x_1,\ldots,x_{d-1})\}.\tag1$$
And lastly, if $M$ is compact, I've seen that people say that $\partial M$ is of class $C^1$ if for each $x\in M$, there is an open neighborhood $\Omega$ of $x$ and a $\psi\in C^1(U)$ with $\psi'(x)\ne0$ for all $x\in\Omega$ and $$\Omega\cap M=\{\psi\le0\}\tag2.$$
How do all these three (the first applied for $\partial M$ instead of $M$) come together? Can we given an equivalent characterization of the second, which does not rely on an appropriate coordinate transformation? And how can we show that if $\partial M$ is of class $\mathcal C$, then $\partial M$ is a $(d-1)$-dimensional embedded $\mathcal C$-submanifold? (I'm willing to assume that $M$ is bounded and open for this implication to hold.)
It is clear that if $\partial M$ is of class $C^1$ (in the sense of the third definition), then $\partial M$ is a $(d-1)$-dimensional embedded $C^1$-submanifold
$^1$ i.e. for each $x\in M$, there is an open neighborhood $\Omega$ of $x$ and a homeomorphism $\varphi$ from $\Omega$ onto an open subset of $\mathbb R^k$ so that $\varphi$ and $\varphi^{-1}$ are of class $\mathcal C$.
$^2$ see Definition 7.2.1 here. I'm not happy with this definition, since it implicitly assumes an appropriate coordinate transformation.
It is possible to have $\partial M$ being a $d-1$-dimensional embedded submanifold without $M$ being class $\mathcal{C}$. In dimension $d=2$, any Jordan curve is ipso facto a 1-dimensional $\mathcal{C}$-submanifold of $\mathbb{R}^2$. Grisvard's Elliptic Problems in Nonsmooth Domains gives a counterexample (Fig. 1.3 and Lemma 1.2.1.4), consisting of a "crumpled" triangle: let $\phi(t):=3|t|-2^{-2k+1}$ for $2^{-2k-1}\le |t| \le 2^{-2k}$ and $\phi(t):=-3|t|+2^{-2k}$ for $2^{-2k-2}\le |t| \le 2^{-2k-1}$, then let $M:=\{(x,y):x>0,\phi(x)<y<x+\phi(x)\}$.
Your third definition looks equivalent to saying that $M$ is a $C^1$ submanifold with boundary of $\mathbb{R}^d$, in which case it is equivalent to $\partial M$ being $C^1$ (in the same reference, after Definition 1.2.1.2).