As far as I know, Tom Sawyer didn’t play a saw, but “sawyer” is a term used by some to refer to people who play the saw. The term usually refers to people who work sawing wood, and according to the Oxford English Dictionary, before that, the term was sawer. I have no idea why someone decided to add a “y” to the word, and honestly, I like it better without it, but no one asked me. In addition to referring to people who saw wood, sawyer refers to people who play the musical saw. We are also called saw players, sawists, and some saw players are even called musicians.

Image from page 994 of “Baltimore and Ohio employees magazine” (1920)