I've been explained that a vector field, when seen as "arrows" in the plane, has 0 divergence when its magnitude doesn't change, i.e. when the "arrows" keep same length. But the following examples puzzle me:
$F(x)=x/|x|$ has always norm 1 but its divergence is not 0
$F(x)=x/|x|^2$ has not constant norm but its divergence is 0
Is there some contradiction or do I have a wrong/incomplete picture?
I don't think your examples contradict your view on the divergence. Take your first example, the divergence is
$$\frac{\partial}{\partial x}\left(\frac{x}{\sqrt{x^2}}\right) = \frac{\sqrt{x^2} - x\frac{2x}{2\sqrt{x^2}}}{x^2} = \frac{\sqrt{x^2} - \frac{x^2}{\sqrt{x^2}}}{x^2} = 0.$$
And similarly we find for your second example:
$$\frac{\partial}{\partial x}\left(\frac{x}{x^2}\right) = \frac{x^2 - x\cdot2x}{x^4} = -\frac{x^2}{x^4} = -\frac{1}{x^2},$$
which is clearly not zero.