Assume we have the projective plane $\mathbb{A}^2=Spec(\mathbb{C}[r,s])$. Now take the projective plane over this affine plane $\mathbb{P}^2_{\mathbb{A}^2}$ with homogenous coordinates $[u:v:w]$.

Define a threefold $Y$ by the vanishing of $u^2+rv^2+sw^2$ in $\mathbb{P}^2_{\mathbb{A}^2}$, i.e. $Y=V(u^2+rv^2+sw^2)$.

On the other hand there is the threefold defined in this question, where its singularities are studied. We have $X=(\mathbb{A}^2\times \mathbb{P}^1)/G$, where $\mathbb{A}^2=Spec(\mathbb{C}[x,y])$ and $G=\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$ with generators $g_1$ and $g_2$. These act via: $g_1.(x,y,[\lambda:\mu])=(-x,y,[\lambda:-\mu])$ and $g_2.(x,y,[\lambda:\mu])=(x,-y,[\mu:\lambda])$.

Are $X$ and $Y$ birationally equivalent? How to test such a thing? If they are, can we write down explicit rational maps $X --> Y$ and $Y --> X$?