Claire,
This is always an interesting topic for me. We did have a few discussions on this over the year on the Forums, but I could not find them for reference. The terms are not always defined the same, but this is the general rule:
As you know, as a general rule, in England they call grosgrain ribbon "Petersham", here we just call it grosgrain. As you say it is a faille weave, with tiny ribs. It was originally made of silk, and you can still find the old vintage or antique silk grosgrain today if you hunt for it. Silk is far superior for millinery uses, but rayon works just fine too. Rayon took over from silk around the early 20th century. I seldom have seen professional millinery ribbon made of cotton but it is out there. Today most of what I find in shops is made of polyester or a rayon acetate blend, neither of which works well for hat making purposes, and the poly is useless in my opinion. Rayon is still available, but not as readily in regular stores but I have not shopped in years in any fabric store. A millinery supply company will have what you need to make a proper hat.
VERY IMPORTANT NOTE: All Petersham/grosgrain ribbon is NOT and I repeat not millinery ribbon! This very important distinction is in the way the edges are bound. While both may look the same to the untrained eye, millinery grosgrain/Petersham (or as we milliners call it...simply millinery ribbon) true millinery ribbon has what we refer to as sawtooth edges on both sides. These look like tiny rounded bumps along the edges, and the edge binding is not straight. This allows the ribbon to take the needed circular shape (when properly pressed and swirled) so it lays perfectly flat against the rounded crown or brim of a hat. If you use the plain type of grosgrain, it will not lay flat and will create gaps that look very unprofessional. Millinery ribbon also allows you to adjust your inside sweatband to the proper hat's size as desired.
Be aware of sellers who call their plain grosgrain ribbon "millinery ribbon". Check the close up photos and if the description does not mention the bumpy edges on both sides, ask the seller before making the purchase.
Claire, I am not sure if I understand your last question. But the true millinery ribbon can be made longer when it is swirled, and I have used it to about any normal size range from small to extra large.