Anti-holomorphic involution of $\mathbb{P}^1$

Question

I wonder if anti-holomorphic involution of $\mathbb{P}^1$ is, up to change of coordinate, given by either $$ z\mapsto \overline{z}, \ \ \,z\mapsto -\overline{z}, \ \ \ or \ \ \ z\mapsto \frac{1}{\overline{z}}, $$ where $z$ is an inhomogeneous coordinate of $\mathbb{P}^1$.