It's so crazy to have credit cards, we call them that here in the EU too but there is usually not much "credit" on them and you're supposed to be on the plus side... What a wild idea.
Credit cards in the US are extremely easy to get and easy to fall into inescapable debt with. I went to a hardware store and signed up for one so I could get 0% financing on some appliances over a year. I asked for like a $2k card. I told them I made $30k a year and they gave me a card with a limit of $8k about 2 minutes later. Now my finances were never even checked, just my credit score. I could have easily lied and said I made way more to get a higher limit, because I actually did lie and said I make less so they wouldn't give me such a high credit limit. It would be very easy for me to get a higher credit than I have income I think.
It almost sounds like you’re describing a charge card instead of a credit card. American Express offers those in the US, but it’s not nearly as widespread in usage as a standard bank credit card.
My credit cards I've had in Germany/Austria were all basically glorified debit-cards which had their own bank account attached to them. Technically I had a credit limit of a couple thousand, but I never went into the negative.
The only difference (for me at least) was that I could use them to rent a car, which is nice.
The way it works here is you can use a normal bank card to pay directly of your checkings account balance, meaning if you have insufficient funds, the card gets declined. A credit (debit) card allows you to pay regardless of you checkings balance. But at the end of the month your negative credit balance must be covered by your checkings account. If it doesn't cover the whole sum, the debt interest kicks in. A true credit card is essentially the same, but instead of automatically balancing the credit, you have to manually transfer the money to the bank. The advantage being that you can use 2 different banks for credit and checking accounts.