They do exist. However "as a convenience" they allow overdrafts at ridiculously high costs. AFAIK you can't shut this feature off and just have charges declined if there isn't enough money in the account.
That completely 100% depends on your bank. I can and have completely disabled overdraft on my debit card. It will reject transactions if my account is empty. (I actually can't imagine a bank not offering to completely disable this, especially if they charge fees for the same. But I'll accept your word on it.)
I thought all overdraft protection is opt-in now? (I thought they passed a law about it in the past few years)
Based on both these replies I went to go look at my account. Apparently the way it works is that you can shut off overdraft protection and if you strictly use the card as debit than any overdraft should be rejected. However if the card is run as credit, which it can be since it has a visa logo, then there's no way to force it to always decline the charge if there isn't enough money in the account.
Banks are really hard to deal with for this kind of stuff. I took the time to get overdraft protection shut off, and a few months later I got a letter from the bank notifying me I'd been signed up for free overdraft protection.