There are wide differences in consumer protections between credit and debit cards.
The list of reasons for a valid chargeback on a debit card is much more narrow than a credit card. For example, suppose you purchase a tablet online and when you get the package, you sign for it, open it, and it's a paperweight. If you charged that on your debit card you're out of luck and will have to go to small claims court. On a credit card -- especially a good credit card -- the charge would be reversed. The issue there is that you signed for the package. If you don't believe that, give your bank's fraud dept a call, and then your credit card companies.
American Express obviously is great for consumer protections but truthfully any Visa Signature card offers a competent level of service.
Personally, I would never use a debit card online (or anywhere else for that matter) and would advise anybody against it. Not to mention, you can earn some fantastic cashback/mileage benefits. My wife and I are flying next spring to Europe, first class from San Francisco, over $20k in airfare, for only $2500 out of pocket for taxes and surcharges. To earn this we've spent $40k between 2 British Air cards in 15 months.