The word towel as a noun is 13th century. What makes you think what you see in the movie isn't a towel? My kitchen towels are plain-weave cotton, though somewhat more loosely woven than a bed sheet (looser weave dries faster). The towels I'm weaving myself are twill, though nowhere near as fancy as 15th century Perugia towels.

I'd say the use to which you put a length of cloth is what determines whether it's a towel or simply fabric you haven't sewn up yet. By that measure, just about anyone of enough means to keep themselves decently dressed should have some sort of towel.

Or is your question "when did terry cloth become the most popular weave for towels?"