For purely online transactions for gadgets purposes, I myself prefer to use Debit Card because you can then control the damage (due to fraud or accidentally charging it).

Try Public Bank Debit Cards (and treat them like prepaid Card), they have branches everywhere to deposit cash. You can open a PBB savings account (not necessary) and link the Debit Card and thus you can pay bills online too. All my children have HLB and PBB Debit cards and got no issue to purchase stuff online. 

If you want a credit card, get Maybankard 2 Cards and you won't have problems using the Visa or MasterCard for your online transactions. The cards are Free For Life.

As for opening online account for CC, not necessary as your will receive monthly statement and will have adequate time to make payment. But if you are familiar with online banking, yah sure, it's FREE to sign up for online account and it is more convenient to make payment.

I myself use RHB Debit Card linked to my savings account for my online transaction and the security is not bad as they will sent a One Time Password to my phone.

Once again, since you are new to credit card, I suggest you click here to read my articles on Introduction to Credit Cards. And from there, you can go read my article on Maybankard 2 Cards.

My children use their cards for ipod/ipad apps, iTunes, ebook, shopping (the girls) and etc. 

Any Visa or MC credit cards should be able to do online transactions unless the card specifically mentioned no online transactions (e.g. Alliance Debit Card) or in some cases the merchant needs a name on the card (e.g. HLB Junior Debit Card and my RHB Debit Card has no name). And I believe your MBB Debit Card also has no name printed on it and maybe that's the reason why you got issue. For example, my children HLB junior Debit Card cannot sign up for itunes account but they can do it with PBB Debit Card.

Try PBB Electron Debit Card (as I said, you can treat it like a Prepaid Card, but there is an annual fee), the card comes with your name on it and you also get a monthly statement.