On $\mathbb{CP}^1$ with standard complex structure, how to prove that there are only two types of antiholomorphic involution, given by $$ \tau :[z:w]\mapsto [\bar w, \bar z] \qquad \eta :[z:w]\mapsto [\bar w, -\bar z]\ ?$$

For $\mathbb {CP}^n$, there should be one involution for even $n$, two for odd $n$.

As suggested: in homogeneous coordinates, let us write our involution as $z \to \overline {Az}$ for some $A \in GL_{n+1}(C)$. Then we have $\overline{A} A = \lambda id$ for some real $\lambda$. By rescaling $\lambda$ and taking determinant, we see that for even $n$ only $\overline{A}A=1$ is possible (class $\tau$ above), while for odd $n$ also $\overline{A}A=-1$ is possible (so also $\eta$ above).