Payment processing acquirer is a descriptive way of saying payment acquirer, and that entry covers the role in full. The extra words appear because, in practice, the firm that acquires a merchant often also processes its transactions, and people want to make clear that both jobs are meant.
An acquirer is a member of a card network that signs a merchant agreement, accepts card transactions on the merchant's behalf, and pays the merchant after deducting fees. Processing is the technical work of carrying authorisation requests, clearing files and settlement data between the merchant, the network and the card issuer. Some firms do both; others split them, so that a bank holds the network membership and carries the settlement risk while a separate payment processor runs the technology. Both arrangements are found in the four-party card model.
The distinction has practical effects. Liability for chargebacks, scheme compliance and merchant underwriting rests with the acquiring member, even if a processor or a payment facilitator handles the day-to-day relationship. When a merchant is asked "who is your acquirer?", the answer is the licensed entity named on its merchant account, not necessarily the brand on its terminal.