The ancestors of this classic trick are commonly believed to be the Nose Bridle (see Scot's Discoverie of Witchcraft, Book 13, Chapter 34, 1584, p. 353.) and an improved version that would become known as Solomon's Pillars (see J. Prevost's La Premiere Partie des Subtiles et Plaisantes Inventions, 1584, p. 76).
The Chinese Sticks, with internal sliding weights, weren't documented until the early 1900s and originated in India, not China. The first explanation of the trick in English seems to be in W. H. J. Shaw's New Ideas in Magic, 1902: “New Hindoo Wands and Cord”. In is unclear if the “new” in Shaw's title refers to the sliding-weight method or Shaw's unusual placement of the cords, issuing from the ends of the sticks rather than through holes in the sides, near one end. If the latter, the title indicates that the trick was already familiar to conjurers.
The next description of the sticks with weights appeared in Stanyon's Magic, Vol. 5, No. 1, Oct. 1904, p. 3. Here, they were called “Devil's Sticks, or Faker's Wands”. Stanyon cites Satya Ranjan Roy of Bhandara, India, as his source of information.
Note that Shaw's use of “Hindoo” in his title, and Stanyon's use of “Faker's” in his, along with his contributing source, all clearly point to India as the origin of the trick. This is further corroborated by Elbiquet in Supplementary Magic, 1917. Writing in India, Elbique calls the trick “Bamboo with String” and places it in a chapter devoted to “A Selection of the Favorite Tricks of Native Jugglers”. The false association of this trick to China seems to have arisen from “Silent” Mora's attribution of it to Hang [sic] Ping Chien in a 1921 advertisement for the trick in The Sphinx, Vol. 20 No. 7, Sep. 1921, p. 257.
For an excellent history of the Chinese Sticks, see Max Maven's “Tracking Slum Magic to Its Lair: the Chinese Sticks” in Gibecière, Vol. 14 No. 2, Summer 2019, p. 137.