I was wondering how could I achieve to have a unique selection from a "border".. Better: How could I switch from a "ring" shaped selection to a "circle" one??
The simplest (and easiest to make accurate) way to get a border selection from a selection is to start with the inner circle, then use
Select > Modify > Border and enter the width you need, in pixels.
This avoids having to start your second selection marquee exactly in the center of the existing selection. It also works for any shape.