I think the problem with your points comes from when you sew the squares together (uppermost two photos). You need to be more accurate at that step. When matching seams is critical like this, I use Elmer's washable school glue instead of pins. I take the squares to the ironing board, match up the seams, then use a straight pin straight up and down to go through both seams at the 1/4" mark, into the ironing board cover. Once the seams are pinned like this, I take the Elmer's glue and place a tiny dot between the seams at each intersection. The glue is inside the seam allowance. Then I use a hot iron to press each seam, which dries the glue, making it act like a "virtual pin". I remove the straight pins before going back to the sewing machine. This is more accurate for me than using pins on the seams.
Oh, and when I need extreme accuracy, such as in this situation, I press all the seams open. I don't try to "nest".