What is the content of Zagiers proof?
What is the actual proof and why does it work? I am not sure I understand why,
there is only one fixed point, and
why that implies that the involution $(x,y,z) \to (x,z,y)$ proves the theorem.
You have $x^2 + 4yz=p$. If the mapping $(x,y,z)\mapsto(x,z,y)$ has a fixed point, then that is a point where $(x,y,z)=(x,z,y)$. That implies $y=z$, so you've got $x^2 + 4y^2=p$. That implies $x^2+(2y)^2=p$, so $p$ is a sum of two squares.
(This is not a complete answer, but it answers at least one of the questions in the posting.)