Ha! I always thought that separate binding was because they didn't leave enough on the backing to do the binding the "right" way. No one ever told me different until I started looking up quilts online.
My grandmother always left extra on the back to bring to the front. It was always to me, stronger than the binding sewn on (we tugged on the edges alot when fighting for the covers)and it wouldn't tear off.
For the last 26 years, my son has had his "tied" quilt and wouldn't trade it for anything. He barely let his daughter use it when she "found" it. He wanted to know why I sewed all over the quilt I made for her and her little sister instead of doing it like his. Some are just traditions that live long in family lines.
If your quilt is made with love (and which one isn't) you made it the very best way that quilt was to be made. Heck with how you did it just that you did it is wonderful.