The Player or Puppeteer is how several of the in-game characters refer to the out-of-game entity that is controlling The Batter and his Add-Ons; you. Others, on the other hand, will simply refer to you as the name you input when beginning the game. Naturally, if you input a name that isn't yours or a bunch of blank spaces, the characters would never know the difference. If you choose not to enter a name, the game will insert "Player" into the field by default.
You also have option to state your gender in the beginning which is also cited several times along with your name throughout the game.