There are two very different types of cards and transactions (in the US) that both go by the name of "debit".
1. As you describe, a card branded with Visa/MC, where the transaction goes through Visa/MC processing. The money is deducted from a checking account, although that fact is immaterial to the merchant.
2. A transaction that goes directly to the bank without Visa/MC. This is the EFTPOS network. This type of transaction is secured by a PIN and the physical card must be swiped (not an online charge). This system was originally for ATM cash withdrawals. But then retail merchants like supermarkets and department stores entered this space for payments, because it's much cheaper for them not to give up the 3% slice to Visa/MC. Wal-Mart in particular was a leader in doing this, and that's why the terminal always prompts for a PIN (it's cheaper for them) rendering it non-obvious how to do a Visa/MC-based transaction instead (unintuitively hit Cancel at the PIN prompt.)
Nowadays, virtually all cards and underlying accounts support both types of transactions. And most customers just know it as a "debit" card, oblivious to the difference between PIN and signature verification. But there are significant differences in the back end.
1. As you describe, a card branded with Visa/MC, where the transaction goes through Visa/MC processing. The money is deducted from a checking account, although that fact is immaterial to the merchant.
2. A transaction that goes directly to the bank without Visa/MC. This is the EFTPOS network. This type of transaction is secured by a PIN and the physical card must be swiped (not an online charge). This system was originally for ATM cash withdrawals. But then retail merchants like supermarkets and department stores entered this space for payments, because it's much cheaper for them not to give up the 3% slice to Visa/MC. Wal-Mart in particular was a leader in doing this, and that's why the terminal always prompts for a PIN (it's cheaper for them) rendering it non-obvious how to do a Visa/MC-based transaction instead (unintuitively hit Cancel at the PIN prompt.)
Nowadays, virtually all cards and underlying accounts support both types of transactions. And most customers just know it as a "debit" card, oblivious to the difference between PIN and signature verification. But there are significant differences in the back end.