Grey Badger, again, is a very continental concept. Grey, in most cases, refers to the brown and gray from Trimmed brushes, Trimmed with a mix of hairs from various parts of the body. Trimmed, too, are more robust, but also scratched as Black Badger, but Trimmed are gentler than Black Badger, again only recommended for veterans or early adopters. Another rare case is Plisson's European Grey, which is a 2-band Badger below.
Pure Badger, of course, literally means Pure Badger hair, but since good quality Badger hair is used on higher grade brushes, Pure Badger is often actually badgers that aren't good enough. Generally refers to a brand of badger hair after HMW-& GT; Silvertip-> After Best's selection path, the remaining badger hair still showed white, brown-white or yellow-brown-yellow color, but its quality was very average, and its toughness was generally lower than that of better quality badger hair, but there was little difference in softness. In addition, some brands may label Black or Grey Badger as Pure Badger, depending on the color and shape of Badger hair. It doesn't matter what it's called, what matters is that you can judge what it is.