As I'm a Taiwanese, I can share something about rice paper. Rice paper is a common name of Chinese paper. Some countries made similar type of paper, e.g. Washi (和紙) in Japan. Rice paper was called "宣紙" in Chinese.
Chinese papers were made by hemp, mulberry, bamboo, or rice straw. "宣紙" first be recorded at Tang dynasty, A.D. 618 ~ 907. The name was from a town called "宣城", means papers made at 宣城. Traditionally, papers made at 宣城 mainly use mulberry bark and rice straw, that's why it be called "rice paper".
Now days, rice paper can be made from many kinds of fibers, even wood pulp. But in serious definition, for ink wash painters and chinese calligraphy, a real rice paper must use barks fiber. Only Moraceae, Ulmaceae and Thymelaeaceae shrubs contain enough cellulose for paper making. Proportion of barks fiber is 40% ~ 80%, others could be wood pulp, cotton or traditionally rice straw and bamboo.
Papers without barks fiber will not be called "宣紙" but "棉紙" in Chinese. The word "棉紙" is cotton paper if you translate directly. But it doesn't mean paper made by cotton. It means the feathery paper edge looks like cotton when tearing the paper.
According to that, Hahnemuhle Rice Paper is actually a Chinese cotton paper, not a real rice paper. Awagami papers are much like rice paper in materials, using mulberry and hemp. Mitsubishi's Pictorico Kenaf is another paper using traditional oriental fibers.