There's a third party between the payment system (website, point of sales systems, card terminals, etc) and the card's provider who also has access to the transaction data in the latter example. These payment systems don't interact with visa/mc/whatever directly.

The tradeoff is you will be giving the merchant your cc info instead, which is not really a good idea if you are dealing with one off transactions as that data is a way weaker link and have more dire consequences when compromised. When it comes to money, someone will always get your info, it's just a matter of choosing who you want to give it to.

Usually there is also online payment providers like Stripe, Adyen, Square, which process the credit card payment and thus get your data - not only the bank and creditcard provider.