Gaussian distribution on a $2$-sphere
Solution 1:
On the circle $S^1$, this is called the von Mises distribution. On the sphere $S^2$, this is called the Kent distribution. There are analogues in every dimension and the two limits you ask for, that are when $\sigma\to0$ and when $\sigma\to\infty$, are as you describe them. This area of expertise is called directional statistics.