And I still have no clue what Movies Anywhere is, sorry.

Think of it like a digital bridge between the big four online video vendors: iTunes, Amazon, Google Play and Vudu. Movies Anywhere is backed by Warner Bros., Disney, Universal, Sony and Fox. So films produced by those studios will appear in the Movies Anywhere app, regardless of where you originally purchased them. Additionally, those films should appear in each of the services through which you've connected your accounts.