It's both a diminutive of Elizabeth and a name in and of itself. Same with Betsy and Liz. Betty can also be a diminutive of Beatrice or Bethia. Betty White is

It's both a diminutive of Elizabeth and a name in and of itself. Same with Betsy and Liz. Betty can also be a diminutive of Beatrice or Bethia. Betty White is Betty and Betty Ford is Elizabeth. There's no "rules" that you can't name someone a name that was traditionally a diminutive of another name, and it's not even particularly uncommon. Alex is common as both a diminutive of Alexandria, a diminutive of Alexandra, and a name of its own. Zach/Zack/Zac can be a diminutive of Zechariah, Zachary, Isaac, or a name in its own right. Max can be short for Maximilian, Maximus, Maxim (which is a shorter form of Maximus), Maxwell, or it can be just Max.