Finite duration vs infinite duration sequences?
There is a slight misunderstanding here. There is no such plane called unit circle plane. Unit circle is in the xy plane (or we call it z plane). The difference is, rather, in the shape of the ROC. For infinite duration sequences, the ROC is in the form of a circular strip in the xy plane (as shown in the second figure).

In the case of finite duration sequences, ROC is the entire xy plane except at z=0 or at z = $\infty $ or at z=0 and $\infty$.

e.g. $x(n) = \{1,\underset{\uparrow}{2},3\} $  

$X(z) = \sum \limits_{-\infty}^{+\infty} x(n)z^{-n}$

$ = x(-1)z^1+x(0)z^0+x(1)z^{-1} = z+2+\frac{3}{z}$

This Z transform will not exist only if $z=\infty$ (for the first term) and $z=0$ (for the last term). This can be generalised for all finite duration signals that their ROC consists of entire z plane (xy plane) except at z=0 or z=$\infty$ or at both.