Lines on a toric cubic surface with a line of nodes

Consider a cubic surface cut out by equations $$x^2y - z^2w$$ inside $$\mathbb{P}^3$$. This gives a cubic surface with a line of nodes, it is toric and has normalisation $$\mathbb{F}_1$$, a Hirzebruch surface.

My confusion is two fold, I am having difficulty spotting which two torus invariant lines of $$\mathbb{F}_1$$ are being glued together.

In addition I would like to understand how this degeneration works with respect to the 27 lines on a cubic surface and see where degenereate to.

** Edit:** Sasha's answer has made me delete some of my post as it has clarified why it is wrong, and now it is unhelpful and long.

The cubic scroll in $$\mathbb{P}^4$$ is isomorphic to $$\mathbb{F}_1$$ and its torus-invariant divisor has three line components and one conic component. The linear projection $$\mathbb{P}^4 \dashrightarrow \mathbb{P}^3$$ from a point lying in the linear span of the conic, but not on the conic itself, gives the normalization map $$\mathbb{F}_1 \to X = \{x^2y - z^2w = 0\} \subset \mathbb{P}^3.$$ Thus, instead of gluing two lines, it collapses the conic onto a line via a 2:1 map.
• I guess with embedding $a^2b - c^2d$ and $bd=e^2$ there should be a way to see this map. – UserUser Nov 18 '19 at 9:15