Spherical coordinates, chart, manifold, differential geometry

I have some doubts about the definition of the chart of manifold applied to the sphere $$S^2$$ embedded in $$R^3$$. In every document I read, the example chart for the sphere is obtained using the stereography projection. No one takes as an example the inverse of the parametrization (for example the sphere of radius = 1)

$$\begin{eqnarray} x = \sin( \theta)\cos( \gamma )\\ y = \sin( \theta)\sin( \gamma )\\ z = \cos (\theta)\\ \end{eqnarray}$$

Why nobody cites this example? Isn't it a chart?

The only reason I may guess is the fact that in order to know the values of $$\gamma$$,$$\theta$$ corresponding to a point on a sphere the center of the sphere is needed and this is an element outside the surface of the sphere. Am I correct?

thank you

In fact, this is a parametrization, and you can use this function on $$\theta\in (0,2\pi)$$ and $$\gamma\in (0,\pi)$$ in order to make this defined on an open set, a requirement parametrizations and charts.