Are credit cards saved to a customer record after a website order?

If the customer does not login first, the order will create a customer record but will not store the credit card. This is considered a guest checkout.
If the customer logs in first, and processes an order, there is an option to Save this information in the Payment Method section of the checkout process. This is checked by default, so the customer would need to uncheck it if they don't want the card info stored.
Reservations booked on the website will create a customer record (if it doesn't already exist) and always store the credit card automatically.