The title says it all really, I searched this website and came across a post with a question titled Why is the Higgs boson spin 0?. But it doesn't really answer my question in the title.
But this next post I found on this site is much more strongly related to my question.
In fact, it is one of the answers to the question in that post which I am essentially questioning. Here is a direct quote for the answer given by @Chiral Anomaly:
The Higgs boson has spin $0$. A photon has spin $1$. The total angular momentum cannot change in the decay, so a Higgs boson cannot decay into a single photon, regardless of the energy. But the total angular momentum of two photons can be zero (because their spins can be oriented in opposite directions), so this decay mode can conserve angular momentum.
The problem with this quote is that it just seems to be just concerned with conserving the spin projection quantum number - which I denote here by $s_z$ and $s$ for the spin quantum number. The $z$-direction in $s_z$ is arbitrary and could be any direction.
As I understand it, photons always have spin, $s=1$. Just like electrons always have a spin of $s=1/2$, but can have $2s+1 (= 2)$ spin projections, namely, $s_z=\pm 1/2$.
So the only way to conserve spin projection would be to have one photon with $-1$ spin projection and the other photon with $+1$ spin projection. Or, have both photons with zero spin projection. That is the only way to conserve spin projection in the $H \to \gamma\gamma$ decay mode. But, the problem is we are not trying to conserve spin projection, but rather spin, $s$, itself.
I tried searching the internet for explanations of why that decay shows the spin of the Higgs boson is different from one, but I cannot get a straight answer. It must be something quite trivial as the original discovery paper in July 2012 simply stated in its abstract that "The decay to two photons indicates that the new particle is a boson with spin different from one" which can be found here after selecting the appropriate title paper which looks like:
So, just to summarize, why does the Higgs diphoton decay prove that the spin of the Higgs is different from one?