The circumcenters of the four triangles of a complete quadrilateral along with the two points of completion form two congruent circles (in black). Surely this must've been done before - what's the formal title for these circles? Update: These two circles are congruent to the Steiner circle (circle $GHIJ$).
Not an answer, but too much for a comment. This is just to point out that there are more of these circles congruent to the Steiner Circle.
What you did for the "opposite" vertices $E,F$ can also be done with the opposite pairs $A,C$ and $B,D$. In the figure, the red circle is what you called the Steiner Circle, and the black pairs of solid, long-dashed, and short-dashed circles come from the three pairs of opposite vertices.
Just messing around I noticed that these circles are concurrent (in threes) at four points which also happen to lie on a circle that is -- surprise! -- congruent with the rest of the circles. And each point of concurrency also lies on a side of the quadrilateral.
As for your actual question .. I've browsed through papers like Clawson's Complete Quadrilateral but nothing jumps out. When doing a literature search, bear in mind that the Steiner circle has other names, such as 'circumcentric' (see Clawson) and 'Miguel'
