Debit comes from a latin verb that means to owe, and credit from a verb that means to trust.

So a debit card is for the money that the bank owes you, and a credit card is for money that the bank trusts you will pay them back.

Sure, which implies the money I store in the bank is debit they owe me. But worded that way I might be induced to think I deserve some kind of interest payment...

It doesn’t just imply it, that is exactly what it is. The bank is loaning your money out and then some (multiplied by whatever the current reserve ratios are). They quite literally don’t have the money you deposited, but just owe it back to you if you ask for it. That’s one of the ways a bank makes money.

It's easy to forget in an era where checking and savings accounts have interest rates so low the average person earns no interest on their cash.

That's the etymological fallacy. Debit and credit just mean left and right side of the ledger.