I would like to know whether there is any nice prescription to give an example of a closed but not exact one-form on $S^2$ (not the $3$-ball). I assume to take some points out of this surface, e.g. 3.
On $\mathbb{R}^2-\{0\}$ a closed but not exact one-form would be $\frac{x dy-y dx}{x^2+y^2}$. Can I use this also for $S^2$?
In general, take a diffeomorphism from $\mathbb S^2\setminus {\rm two\ points}$ to $\mathbb R^2\setminus\{0\}$ and pull back the form that you have there.
If for example the two points that you remove are $(0,0,1)$ and $(0,0,-1)$ then you can write every point as $(r\cos\phi,r\sin\phi,z)$ and $\phi$ is well-defined up to multiples of $2\pi$. Hence $d\phi$ is a well-defined one-form which does what you want.
Indeed the description in the previous paragraph is unnecessarily complicated. If you work it out you arrive exactly at the form $\frac{xdy-ydx}{x^2+y^2}$ that you mentioned in the question. Of course, $x=r\cos\phi$, $y=r\sin\phi$.