As someone who leads a lot of historical dance, especially eighteenth century, I think of "quadrille" as referring specifically to the dance format in square formation that replaced eighteenth century cotillions. There are distinct structural differences between cotillions, quadrilles, and most modern square dances. I'll spare you the details, at least for now. Although there are plenty of modern square dances with the word "quadrille" in the name, I don't think of them as quadrilles.
Jacob Bloom
Arlington, Massachusetts