Computing Cartier Divisor from Weil Divisor Example

I am trying to work through the following problem:

Let $k$ be a field, and let $X = \operatorname{Spec} k[x, y, z, w]/(xy−zw)\subseteq \mathbb{A}^4_ k.$

(a) Show that $D= V (x, z)$ is a prime (Weil) divisor on $X$ and that $\operatorname{Cl} X\simeq \mathbb{Z}$ is generated by the divisor class of $D$. [Note: you may assume that $X$ satisfies the hypotheses for the definition of class group.]

(b) Let $Y= X \setminus\{{O\}}$ where $O$ denotes the origin in $\mathbb{A}^4_ k$. Show that $D_Y := D ∩Y$ defines a Cartier divisor on $Y$, and describe the line bundle $O_Y (D_Y)$ via transition maps on a suitable open cover of $Y$.

Now part (a) is easy but I am not sure why my attempt for (b) is not working. Clearly $Y$ has an open cover given by $D(x),D(y),D(z),D(w)$. On $D(x)$ say the ideal $(x,z)$ becomes trivial in the local ring so I think I want representing rational function $f_x=1$. For a more interesting case on $D(y)$, we have identity $x=zwy^{-1}$ and so the ideal becomes $(z)$ in the local ring.

Now clearly on $Y$, the divisor $(z)=V(x,z)+V(y,z)$ by checking that z is a unit on the complement of this, but by restricting to $D(y)$ we get $(z)=V(x,z)$.

My problem then occurs in checking this actually is a Cartier Divisor, on the overlap $D(xy)$, $z$ is not a unit. So clearly something is wrong I am just not sure what ? Any help would be appreciated.