Credit Card
While hardly anonymous, credit cards are a very convenient way to pay.
If this payment method is chosen, I will provide you with the link to my wishlist where the appropriate gift can be selected.
The statement descriptor on your credit card bill may vary depending on the bank/cc, but it generally will say some version of the wishlist company name. There will be no details about the gift title or recipient.
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Let me know if you have any questions.