I'm curious about how the credit cards work.

When you do a bank transfer it is you who tells your "bank account" what amount to send and where to send it. When you pay with credit card it seems reverse. You give all required card details to merchant so he can take from your account the billed amount.

Is this how it works? I always thought that this is the case but I talked recently about this process with someone and it seemed crazy to him. He is sure that the dollar amount is also part of the process (meaning the bank would allow only this amount to be charged by mechant) and that merchants coundn't technically be able to charge more. Isn't the sale amount just some number that is simply entered to some payment processing machine but is not part of the specific transaction from technicall POV like with the bank transfer example and ergo merchant technically can charge whatever amounts he wants regardless of "online website cart amount" when confirming payment?