Harmony, I shouldn't have even mentioned my blanket-stitch binding because that is my attempt at a one-step machine binding, so it's different. I sew the long strip edges together lengthwise, then press placing the raw-edge seam in the middle of the side that will be hidden anyway, wrap it around the edge with slightly more fabric on the back than front, and apply from the front with a machine blanket stitch. Only the "bites" the blanket stitch is taking every 8th-inch or so are really entering the front of the binding, because I'm guiding it right along the edge, so it helps it drape nicer. When you attach a one-step machine binding with a straight stitch, the quilt edges have a tendency to "flutter" rather than drape straight. My basket quilt flutters, for sure.
For hers, she started out as you would to apply a separate binding that you are going to turn and hand-stitch. But for the second step instead of hand-stitching, when she turned it to the back, she sewed using that "S" curve machine stitch. I'm guessing she was looking at the back of the quilt as she sewed. The binding was somewhat flattened, but very neat.
It's "muslin". There are a lot of different weights and "feels" of it. I bought some one time that was so soft but never have been able to find any that soft again. It made a perfect baby quilt backing.