Credit card numbers are much safer to use online than debit cards, mostly because credit card dispute mechanisms give consumers a lot more power. If a merchant behaves in an unsavory way, with a credit card you can usually just call your bank, issue a chargeback and that's the end of it. You can't do that with a debit card.
You absolutely CAN dispute charges with debit cards. The problem OP might have with debit cards could be that a fraudulent charge temporarily locks cash funds (with debit card) instead of credit availability (with credit card).
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.
You can dispute most debit card transactions now.
Besides the "one is availability of credit, the other, real money", there is a liability difference.
It used to be, at one point, that debit card liability was essentially unlimited but a consumer's liability for fraud on their credit card was limited to $50 by federal law (see 15 USC § 1643). Debit card liability now has a maximum as well, but your maximum liability actually depends on how quickly you notify.
If it's within two days, it's $50 liability on a debit card, but if you notify past that, they could legally make you liable for up to $500.
(There are other liability generating situations that exist for debit cards but not credit cards)