What is the name of the process that describes what happens when a customer is taken to a separate site to complete a transaction before being sent back to the original site again?

Third-party payment processing is correct because it accurately describes the scenario where a customer is redirected to an external service (the "third party") to handle the financial transaction. This service processes the payment securely and then returns the customer to the original merchant's website. This method is common for e-commerce sites that don't want to directly handle sensitive payment information themselves, relying instead on specialized payment gateways. Second-party payment processing is not a standard industry term for this process. External payment processing is a broader, less specific term that could describe various methods, but "third-party" specifically denotes the involvement of an independent entity for the transaction.