The Thumb Tie appears to be the invention of Giuseppe Pinetti (1750–1800). Henri Decremps explains two methods for it in Supplément à la magie blanche dévoilée, 1785, p. 24. Henry Ridgely Evans described Pinetti’s presentation in The Sphinx, Vol. 1 No. 2, Apr. 1902, n.p.
The trick is frequently attributed to the famous Japanese magician, Ten-ichi (1853-1912). He openly used two cords for the binding, and a different method for gaining slack. His presentation was original and quite different from that done by Western magicians. However, there is evidence that strongly suggests the effect was brought to Japan from the West. See Mitsunobu Matsuyama's study in Gibecière, Vol. 5 No. 1, Winter 2010, p. 95. Ten-ichi's method and presentation are described by Harlan Tarbell in the Tarbell System, Lesson 11, 1926, p. 10.