Birational permutations of the projective plane. Another point of view

A year ago, the proof of the following theorem was discussed at a seminar. Let $q=2^m$ and $q>=4$, then birational permutations of the projective plane induce only even permutations of $\mathbb{F}_q$ - points of the projective plane. The idea of the proof was to explicitly describe the generators of the group of birational permutations, and prove for each generator that it induces an even permutation of rational points.
This time we will prove the mentioned theorem in a completely different way. Namely, following the article by A. Genevois, A. Lonjou and K. Urech, using some technique, we generalize the concept of parity to the entire group of birational automorphisms and show that all elements of finite order (in particular, involutions) are even elements. We will also see that this approach allows us to generalize the theorem to an arbitrary smooth rational projective surface.
If there is time left, we will discuss what other statements and theorems can be proved using a similar technique.