Chiburi is a modern reading of a word that appears in the densho of Eishin-ryu as either 血振 or 血震. The original pronunciation is most likely chiburui, which is the reading you find if you look the word up in a Japanese dictionary such as Iwanami Shoten’s Kojien. In his book Koryu Iai no Hondo, the late Iwata Norikazu quotes another Eishin-ryu teacher, Morita Tadahiko, as being correct in his assertion that “chiburui” is the accurate term and that “chiburi” is in fact a mistaken reading (the word “chiburi” that appears in the dictionary actually refers a method of preparing fish).
Most beginners learning iaido will be taught that the motion of chiburi is intended to fling the blood from the tip of the sword after cutting. In most books on iaido too, chiburi is described as serving this purpose. Muso Jikiden Eishin-ryu and Muso Shinden-ryu also contain chinugui (wiping the blood from the blade with a cloth, paper or the fingers) in a small number of techniques in the first teaching level of Omori-ryu (Shoden/Seiza no bu). In Muso Jikiden Eishin-ryu at least, this is technically done by putting one hand inside one’s hakama and using that to wipe the blade. In practice however, the shape is performed but the blade is not really wiped on the hakama. According to Mori Shigeki, this is because this because the oil used on swords in Oe sensei’s day would soil the clothes.
Despite more people becoming aware of it recently, the idea that chiburi isn’t really a practical method of removing blood from the blade is not recent – it has been expressed by teachers in Japan for a long time.