I have interpreted the word credit card to encompass all credit and debit cards. While I have a credit card I never use it as for any on-line payment I have ever had to make, a debit card is acceptable. I am assuming that debit cards will be acceptable and that there are few religions, if any, which have objections to them.
If you are a resident of the UK, it's actually quite a lot safer to use a credit card online than a debit card.
This is because of the extra protection you automatically get from the Consumer Credit Act when you use a credit card.
Reversing a debit card transaction is relatively difficult and takes a while, during which time you may risk overdraft charges etc which are not always possible to recover.
- It took a few weeks for me to get the money back after somebody nicked my wallet and used my debit cards, and I never got all of it.
Clearing up a Direct Debit error is slow and relatively painful from the twice that happened.
I've never tried reversing bank transfers, but my mum tells me it's much harder than DD.
Credit cards will normally 'freeze' the transaction immediately pending investigation, you rarely need to contact the merchant at all, and you usually have a month between statement and owing the actual money. You can claim quite a long time after the transaction - and even for things like "goods not fit for purpose".
The few fraudulent credit card transactions I've suffered were resolved to my satisfaction long before the bill became due, so I was never left out of pocket.
Pre-paid "credit" cards worry me though, as I have no idea if they are actually covered by the CCA - possibly not.
Other nations have different laws of course.
That said, I know some people do have a lot of trouble controlling their spending with credit cards.
I treat mine as a "debit card plus" - I only want the CCA protections, I don't want to actually 'borrow' anything so I always pay the whole thing every month.