Let's call your coordinates $\phi$ and $\theta$, as is more usual. Thus $(x,y,z)=(sin \phi\ cos\theta, sin\phi\ sin\theta, cos\phi)$. 

Yes, in the centralizer of $SO(2)$ there is the group that leaves $\theta$ unchanged, $(\phi, \theta)\mapsto (f(\phi ),\theta)$ where $f$ is a diffeomorphism from $[0,\pi]$ to itself that is nice enough at the endpoints. Also there is the group that leaves $\phi$ unchanged, $(\phi, \theta)\mapsto (\phi ,\theta +g(\phi))$ where $g$ is a smooth map from $[0,\pi]$ to $\mathbb R/2\pi \mathbb Z$. The centralizer is the semidirect product of these, and the normalizer is bigger by a factor of two.