I guess I stand corrected!
So suppose a person writes an app that uses the Microsoft framework for handling in-app purchases. Is that then billed through the carrier, or does it provide an option for different payment methods when making the purchase?
It depends - if carrier billing is available, then you would get the option to pay that way, or via CC/DC through Microsoft. Similarly on Android & iOS, you could pay via Google Wallet/App Store billing, or via carrier if the choice is there. If no carrier billing, then the answer is via WP Market, Google wallet or App store.
The availability of carrier billing depends on both the handset AND the carrier. So you cannot assume that all WPs in a certain country will have it available, for example. Even for all WP on a given carrier, you may have to choose a certain plan or activate the option.