Regarding the "proper binding" vs "the fold over thing", as with most things, there will be more ways to finish a quilt than any of us will likely ever attempt. The edges of quilt seem to take a lot of wear and the way it is finished doesn't change that...but some seem to be easier to replace than others. To facilitate replacement of worn bindings, I've known some to use 4 piece bindings...one for each side.
The "fold over thing" will likely wear sooner if it is but one layer of fabric at the edge...and likely does not have the stay stitching that an applied binding will create if machine sewn one side and hand sewn on the other. Bias binding keeps the wear from being on grain...but straight of grain binding can be cut just slightly askew and accomplish nearly the same goal. It's always the quiltmaker's choice...I'm not in favor of restrictions of any kind... These differences are some of what I celebrate about our craft. Hope this helps.