Let $X$ be a variety over $\mathbb C$ and $\pi: X\to \Delta$ be a flat morphism over the unit disk $\Delta=\{z:|z|<1\}$. Let $Z$ be a component of $X_0=\pi^{-1}(0)$. The multiplicity of $Z$ is defined as the order of vanishing of $\pi^*(t)$ on $Z$, where $t$ is the local coordinate of $0\in\Delta$.
I'm trying to understand the multiplicity through a complex analytic point of view, via the following way.
Let $p\in Z$ be a general point (in particular, $p$ is a smooth point of $Z$ and does not meet other components of $X_0$). Let $\Delta_p$ is a holomorphic disk in $X$ which is transversal to $Z$ at $p$. Then the restriction
$$\pi|_{\Delta_p}:\Delta_p\to \Delta$$
is identified with $z\mapsto z^m$ for some $m\ge 1$.
Question: Is it true that $m$ coincides with multiplicity of $Z$ in general?
My motivating example is the family of plane curves $X=\{y^2-tx=0\}\subset \mathbb C^2_{[x,y]}\times \Delta$, take $p=(1,0,0)$ and $\Delta_p=\{(1,t,t^2):t\in \Delta\}$ which is 2-to-1 to $\Delta$, so the degree coincides with the multiplicity of $\{y^2=0\}$. Also the degree is not dependent on the choice of $p$ (except at $p=[0,0,0]$ where the total space is singular) and choice of normal disk.
This leads to the construction in my question as above, and I think the answer should be true. However, I haven't seen such a complex analytic description in any reference before. Is that true? I would appreciate it if anyone can show me proof or reference.
p.s., My original question was posted on MSE here but I received no comment/answer so far.
