Can you explain to me the relationship between those two definitions of branched covering on surfaces?

Question

I'm studying branched (or ramified) coverings between surfaces and the definition that was given in the book I'm reading is the following:
"Lets consider two closed and connected surfaces $M$ and $N$, we say that the fuction $f:M\longrightarrow N$ is a branched covering if there exists a finite set of points {$x_1,...,x_n$}$\subseteq N$ such that the set $f^{-1}(${$x_1,...,x_n$}$)$ is discrete and the restriction of $f$ to the set $M\setminus f^{-1}( ${$ x_1,...,x_n$}$ )$ is a topological covering."
Then I read that every branched covering is locally modeled on functions of the form $z\mapsto z^m$ and this can also be used as a definition for branched coverings.
I just can't find a proof for that, and I don't know where I could get it. Any advice/reference?

EDIT:
I see that the second "definition" is not so clear, that's because I actually didn't find a formalization of it, but I think it could go like this:
"Given two closed and compact surfaces $M$ and $N$, a function $M\longrightarrow N$ is a branched covering if it is "locally modeled" (I didn't even find any definition of what being locally modeled is, however I tried formalizing it below) on functions of the form $z\mapsto z^m$ where $z\in\mathbb{C}$ and $m$ is a positive integer. For any point where $m>1$ the point corresponding to $0$ in the target $\mathbb{C}$ is called a branching point. The branching points must be finitely many".
For the definition of "being locally modeled", I think it could be something along the lines of:
"Given two surfaces $M$ and $N$ and given a continuous function $f:M\longrightarrow N$, lets consider a point $y\in M$ and its image $x=f(y)$. If there exist:

1. Two open neighborhoods $V,U$ of $y$ and $x$ respectively
2. Two homeomorphisms $\varphi:V\longrightarrow D$, $\psi:U\longrightarrow D$ such that $\varphi(y)=0$ and $\psi(x)=0$ and $D$ is the unitary open disk centered on $0$
3. A function $p_y:D\longrightarrow D$

Such that $f\circ\varphi^{-1}=\psi^{-1}\circ p_y$, then we say that $p_y$ is a local model of $f$ in $y$".
If you have a better definition than this, that maybe just captures the idea of being "locally modeled" on $z\mapsto z^m$, it would be appreciated.