When you use a debit card and process it as credit, it's charging it directly to the bank/credit union who is issuing the card. Some banks/credit unions charge a fee for using it as a credit card, but those fees are charged to the cardholder and not the company.
Debit card - use as credit - I pay a fee from my bank out of my own pocket
Credit card - Target pays a fee to process the card through the payment company