Back in $1895$, before there was the notion of topology, Poincaré had written his definition of manifolds in Analysis Situs. The definition (in French) can be found here. Stated with modern symbols, it goes like this:
($\star$) A nonempty subset $M\subseteq\mathbb{R}^n$ is a manifold of dimension $n-p$ if there exists $F\in C^1(\mathbb{R}^n;\mathbb{R}^p)$ and $\varphi\in C^1(\mathbb{R}^n;\mathbb{R}^q)$ such that $\operatorname{rank} J_F(x)=p$ for all $x\in M$, and that $M=\{x\in\mathbb{R}^n:F(x)=0,\varphi(x)>0\}$.
Here $J_F$ is the Jacobian matrix of $F$, and $\varphi(x)>0$ means each entry of $\varphi(x)$ being positive.
For me, I would like to state this definition without using inequalities as
($\star'$) A nonempty subset $M\subseteq\mathbb{R}^n$ is a manifold of dimension $n-p$ if there exists $F\in C^1(\mathbb{R}^n;\mathbb{R}^p)$ and an open set $O\subseteq\mathbb{R}^n$ such that $\operatorname{rank} J_F(x)=p$ for all $x\in M$, and that $M=\{x\in\mathbb{R}^n:F(x)=0\}\cap O$.
The implication ($\star$)$\Rightarrow$($\star'$) is clear; for the reversed implication, take a smooth function $\psi:\mathbb{R}^n\to\mathbb{R}$ such that $\psi^{-1}(0)=\mathbb{R}^n\setminus O$ (see here), and taking $\varphi=\psi^2$ will do.
So a natural question is: is Poincaré's definition equivalent to the one now we are used to, which is (I'm copying the definition stated in this question):
A nonempty subset $M\subseteq\mathbb{R}^n$ is called an $m$-dimensional submanifold of $\mathbb{R}^n$ if for every point $x_0\in M$ there exists an open set $U\subseteq \mathbb{R}^n$ containing $x_0$ and an open subset $V\subseteq \mathbb{R}^n$ together with a $C^1$-diffeomorphism $\varphi$ from $U$ to $V$ such that $\varphi(M\cap U)=V\cap(\mathbb{R}^m \times\{0\})$ with $0\in\mathbb{R}^{n-m}$.
Of course what satisfies Poincaré's definition satisfies also the modern definition: for every $x_0\in M$, their exists $\{i_1,\cdots,i_p\}\subset\{1,\cdots,n\}$ such that $\det\left.\dfrac{\partial F}{\partial(x_{i_1},\cdots,x_{i_p})}\right|_{x=x_0}\neq 0$. Consider $\varphi$ defined on a neighborhood of $x_0$ that is included in $O$ as $$ \begin{cases} (\varphi_1,\cdots,\varphi_p)(x) = F(x),\\ \varphi_{p+1}(x) = x_{j_1},\\ \cdots\\ \varphi_n(x) = x_{j_{n-p}}, \end{cases} $$ where $\{j_1,\cdots,j_{n-p}\}=\{1,\cdots,n\}\setminus\{i_1,\cdots,i_p\}$, then $\varphi$ is a $C^1$-diffeomorphism from a neighborhood $U$ of $x_0$ that is contained in $O$ to an open set $V\subseteq\mathbb{R}^n$, and $\varphi(M\cap U)=V\cap(\mathbb{R}^m \times\{0\})$.
What about the other direction? Can every submanifold $M$ of $\mathbb{R}^n$ be written as the intersection of the zero set of a $C^1(\mathbb{R}^n;\mathbb{R}^p)$ function and an open set, where the function is require to be regular on $M$ (Jacobian matrix having full rank at every point on $M$)?
Thank you for your help.
