According to wikipedia, which is extensively sourced on this topic (although I haven't checked any of them myself for reliability), says
The earliest Japanese "sansukumi-ken" game was known as "mushi-ken" (虫拳), which was imported directly from China. In "Mushi-ken" the "frog" (represented by the thumb) is superseded by the "slug" (represented by the little finger), which, in turn is superseded by the "snake" (represented by the index finger), which is superseded by the "frog". Although this game was imported from China the Japanese version differs in the animals represented. In adopting the game, the original Chinese characters for the poisonous centipede (蜈蜙) were apparently confused with the characters for the "slug" (蛞蝓).
Today, the best-known "sansukumi-ken" is called "jan-ken", which is a variation of the Chinese games introduced in the 17th century. "Jan-ken" uses the rock, paper, and scissors signs and is the game that the modern version of rock-paper-scissors derives from directly.
Its English-language name is therefore taken from a translation of the names of the three Japanese hand-gestures for rock, paper and scissors: elsewhere in Asia the open-palm gesture represents "cloth" rather than "paper". The shape of the scissors is also adopted from the Japanese style.
7) Linhart, Sepp (1995). "Some Thoughts on the Ken Game in Japan: From the Viewpoint of Comparative Civilization Studies". Senri Ethnological Studies 40: 101–124. hdl:10502/750.
8) Linhart, Sepp (1995). "Rituality in the ken game". Ceremony and Ritual in Japan. London: Routledge. pp. 38–41. ISBN 9780415116633.
9) Sosnoski, Daniel (2001). Introduction to Japanese culture. Rutland: Tuttle. p. 44. ISBN 9780804820561.
12) 長田須磨・須山名保子共編 (1977.4). 『奄美方言分類辞典』上巻. Tokyo: Kasama shoin. ASIN B000J8V5WU.
Assuming that the sources are correct, there's no truth to this legend, because the "paper" name wasn't introduced until long after the game had spread out of China.