What’s the difference between a settlement bank and a remitting bank?

Remitting bank in most cases is the originating bank, the institution that is originating the transaction on behalf of their customer.

The settlement bank is the one that is the designated bank (usually in a country) or usually for a particular transaction type (example credit card payments settlement for say MasterCard), which would initiate the payment for further (and hopefully final) credit and provide back the confirmation of the payment, hence closing the transaction loop.