The typical definition is: for a holomorphic function $f$ vanishing at 0 with order $d$ if exists a neighborhood of $0\in \Bbb{C}^n$ with the nonvanishing Taylor coefficient being $d$ inside that neiborhood.
There is another one, we can restrict $f$ onto line passing through the origin, and it becomes a single variable holomorphic function, in this case we can also define the order of vanishing
I want to show order of vanishing of $f$ is the minimal of the order of vanishing restrict to the line.
if it's infinite, there is nothing to prove, as implication of indentity principle.
if it's finite, I don't know how to prove it.
That's not so hard, the key observation is that all the derivatives restricted on a line can be linearly expressed in terms of it's partial derivative.
let me denote $m =\inf_v\{\text{ord}_0\ f|_{v}\}$
As inf of set of integer is always attainable, denote it $v$, hence $g(z) = f(zv)$ as a holomorphic function has order $m$ i.e. locally the partial derivative of order m does not vanish for $g$. However, this expression is attained by combination of partial derivative of $f$, it implies there is some partial derivative of $f$ with order $m$ does not vanish, therefore $\ \text{ord}_0\ f \le m$ however as we assume $m$ is Infinium for all direction in particular for coordinate direction, it implies all the coordinate direction has vanishing derivative upto oder $m-1$, therefore $\ \text{ord}_0\ f \ge m$.