Please allow me to add to your confuzzilation.
The square in your diagram is ON POINT - so the DIAGONAL of the square (point to point) is what would be be 6" FINISHED, not the sides of the square - they would need to be smaller.
I understand what you're saying, but that's not how I'm doing it (or rather, did it). Rather than cutting the center to exactly its size, I used the snowball method, where you make the back piece the full size (6.5") then sew stuff on top of it. So it did need to be the 6.5". I see how to calculate it with the way you're talking about it, with the dividing by sqrt 2 (which is approx 1.414), but I wanted to deal with a full sized finished piece, rather than bias edges on the wings + matching up with my approximated square