Working with Morphisms in Local Coordinates

I don't know how much this will help you in general but here is how I would deal with your particular example :

Put $dx = x_1 - x_0$ and $dz = z_1 - z_0$, so you have a problem at $dx(dx+dz)=0$. Then write $$X(- \mu) = \frac{dz^2-dx(z_0dx-x_0dz)}{dx(dx+dz)}- x_0-x_1 $$ You can see that $X(- \mu)$ (as well as $Z(- \mu)$) is homogeneous of degree $0$ in $dx,dz$

From your equation $z+z²=x^3+zx²$, with a process analogous to differentiation, you get $dx(x_1^2+x_0x_1+x_0^2+z_0x_0+z_0x_1) = dz(1+z_1+z_0-x_1^2)$

This allows you to replace $dx$ with $1+z_1+z_0-x_1^2$ and $dz$ with $x_1^2+x_0x_1+x_0^2+z_0x_0+z_0x_1$ in your expressions, and you obtain a rational function with denominator $dx(dx+dy) = 1 + \ldots$ for $X$ and $dx^2(dx+dy) = 1 + \ldots$ for $Z$, so they are valid on a neighboorhoud of $(0,0) \times (0,0)$.