How to obtain the following topology? My follow up question: How can I use a pivot circle instead of a pivot point

Question

At this moment, I have this:


What is the most efficient way to achieve a quad as cut section, where


all the faces as well as the unselected vertices are duplicated and
overlapping
but share the same vertices (selected).


The cut section would look like this:
Unless I've severely misunderstood what you're trying to achieve, I still think a torus with Minor Segments set to 4 is the easiest and quickest way.

ShiftA followed by M followed by T or Add -> Mesh -> Torus to add the torus.



After adding it and changing Minor Segments to 4, just scale along Z, S followed by Z, then type in the scale or drag the mouse, and finish with Enter or LMB.