Q: How do I delete a debit or credit card from my phone

Do the instructions on the page that you posted from not work (your payment details are tied to your iTunes account, not the device) ? If you aren't getting the 'none' option do any of the scenarios on the 'if you don't see None or can't select it' link on that page apply ?

You won't get the 'none' option, the red text at the top of that screen means that iTunes wasn't able to collect the money for your last purchase and that you therefore owe iTunes money - that will have to be paid off (if your current card isn't being accepted do you have another card to try ?) before you will potentially get the 'none' option, and you also won't be able to download anything else until it's paid off.

Paying a debt : Pay an unpaid balance in the iTunes Store - Apple Support