The Scarf debate - Echarpe v. Foulard?

So this debate started one day in my class at FIAF. I am a woman of the scarf, silk, many vintage, that I wear nearly every day, especially with my boring lawyer suits. We were discussing something that related to shopping and I said something about the foulard I was wearing, a large square silk printed scarf. Perhaps a Celine or Leonard, Longchamp or Lancel. The designer doesn't matter for this discussion. My professor corrected me and suggested that it was an echarpe, and that foulard is only for covering the head.

Now, I remember quite a few years ago in Paris with D. shopping in my favorite vintage store, Reciproce, where I was looking at a beautiful Celine scarf that was calling my name and saying "take me home to America." I was already wearing another silk scarf. The sales woman turned to my husband and we were discussing the scarf and told D. that I was a woman of the scarf "la femme de le foulard". Now here was a fashionable Parisian who knew the difference between foulard and echarpe.

Naturally, being me, since this discussion I have been researching the subtle differences between the two terms. What it appears to be is that a foulard is decorative, a square with a pattern for fashion. It can be a head covering. While an echarpe is usually wool or a heavy material and used for warmth.

If you have a different explanation, comments are welcome as we are always learning and open to information for sharing.