I'm trying to understand the relationship between the different models of del Pezzo surfaces of degree $2$.

Let $k$ be a field of characteristic not equal to $2$. Usually, del Pezzo surfaces of degree $2$ are considered as hypersurfaces of degree $4$ in $\mathbb{P}(1,1,1,2)$: $$S: \quad w^2 = f(x,y,z)$$ with $\deg f = 4$.

There are however other natural models, such as hypersurfaces of bidegree $(2,2)$ in $\mathbb{P}^1 \times \mathbb{P}^2$: $$q(s,t,x,y,z) = 0,$$ where $q$ is bihomogeneous of degree (2,2) in $(s,t)$ and $(x,y,z)$, respectively.

Call these models of type $1$ and $2$, respectively. I'm trying to understand how one goes between these different models.

Given a surface of type $2$, how does one construct a model of type $1$ (i.e. a model in weighted projective space)?

I know how one *should* go about doing this. The anticanonial class $-K_S$ is the fibre of the projection onto $\mathbb{P}^2$. So a basis for the anticanonical sections is $x,y,z$. Now the space $H^0(S, -2K_S)$ is $7$-dimensional, so there should be a section $w$ in here which is linearly indenpendent from the obvious sections $x^2,xy,xz,y^2,yz,z^2$. With this in hand, we obtain the required map
$$S \to \mathbb{P}(1,1,1,2), \quad (s,t) \times (x,y,z) \mapsto (x,y,z,w).$$
But what is $w$? I don't see how to determine $w$ explicitly.