Miranda's book has more of a complex geometry flavor, so he defines order of vanishing using holomorphic charts and local coordinates. (It's really not that different from using uniformizers and local rings, though.) The analogue of the result you're looking for is the passage preceding Theorem 2.3 of Chapter I (p. 11), which says more or less the following.
Proposition. Let $C$ be a nonsingular affine plane curve given by the equation $f(x,y)=0$, where $f \in \mathbb{C}[x,y]$ is irreducible. Given $P \in C$, if $\frac{\partial f}{\partial y}(P) \neq 0$ then $x$ is a local coordinate on some neighborhood $U$ of $P$. In other words $(U, \pi_x)$ is a holomorphic chart containing $P$, where $\pi_x : (x,y) \mapsto x$.
(And similarly if $\frac{\partial f}{\partial x}(P) \neq 0$, then $y$ is a local coordinate near $P$.) The proof is basically the Implicit Function Theorem (Theorem I.2.1). Since the tangent line at the point $P = (a,b)$ is $\frac{\partial f}{\partial x}(P) (x-a) + \frac{\partial f}{\partial y}(P) (y-b)$, this agrees with the criterion you stated in terms of tangent lines.
Okay, so let's apply the proposition to your example. Let $E$ be the projective curve given by $Y^2 Z = X^3 - X Z^2$. (Forgive me for changing the notation, but I prefer to use lower case letters for affine coordinates.) In the affine open where $Z \neq 0$, $E$ is given by $y^2 = x^3 - x$, where $x = X/Z$ and $y = Y/Z$, so let $f = y^2 - (x^3 - x)$. Here $p_2 = (1,0)$, and since $f_x(p_2) = -(3(1)^2 - 1) \neq 0$, then $y$ is a local coordinate near $p_2$, so $\DeclareMathOperator{\ord}{ord} \ord_{p_2}(y) = 1$. (As you said, its Taylor series is simply $y$!)
To compute the order of vanshing at $p_0 = [0:1:0]$, we move to the affine open where $Y \neq 0$. Letting $w = X/Y$ and $z = Z/Y$, then $E$ is given by $z = w^3 - w z^2$, and $p_0 = (0,0)$. From this equation we can already see that $z$ can't be a uniformizer at $p_0$ since $z = w^3 - w z^2$ and both terms on the righthand side vanish to at least order $3$. Letting $h = z - (w^3 - wz^2)$, then
$$
h_z(p_0) = 1 - (-2wz)\big{|}_{(w,z) = (0,0)} = 1 \neq 0
$$
so $w$ is a local coordinate near $p_0$. To compute $\ord_{p_0}(z)$ we need to express $z$ as a Taylor series in $w$. One way to do this is by repeatedly substituting $z = w^3 - w z^2$ into the righthand side of the equation:
\begin{align*}
z &= w^3 - w z^2 = w^3 - w (w^3 - w z^2)^2 = w^3 - w^7 + 2 w^5 z^2 - w^3 z^4\\
&= w^3 - w^7 + 2 w^{11} - 5 w^{15} + \cdots
\end{align*}
If you don't like that approach, the Implicit Function Theorem implies that $z = g(w)$ for some function $g$ holomorphic near $w=0$. Substituting this into the equation for the curve yields
$$
g(w) = w^3 - w \, g(w)^2
$$
and we can compute as many Taylor coefficients as we like by repeatedly differentiating and substituting $w = 0$.
In any case, we find that
$$
\ord_{p_0}(y) = \ord_{p_0}(1/z) = -\ord_{p_0}(z) = -3 \, .
$$
