Bilinear Form: Weil pairing
See page 17 of the thesis you linked to. There he mentions affine coordinates and homogenization. For your points, $Z=1$. So just set $Z=1$ in your functions above.


  To find the poles, we must remember that as a rational
  function, $Y = Y/1$. Homogenizing, we have $f(X, Y, Z) = Y/Z$. Then to find the poles
  of $f$, we must analyze $Z$ as a polynomial. For all affine points on $E$, $Z = 1$, so $Z = 0$
  only at $\mathcal O$.


Here's more about projective coordinates.

Keep in mind that in general, the sum of two points in projective coordinates having $Z=1$ is not a point with $Z=1$. In this case, the author already did the sums $P_3 + S$ and $P_1 - S$ in affine coordinates.