"I pay with credit card" is rejected as an answer in English, but I think it should be accepted. As far as I know "pay with" and "pay by" are interchangeable and mean the same thing. (A very unscientific test of Google searching "pay by" and "pay with" returns more results for the latter, in fact).
However, our purpose here is to teach proper Greek, not common colloquialisms. Emphasis on Greek, as this should be everyone's goal when taking this course. To learn or practice Greek, not search for common colloquialisms in English :P
Sure, there are lots of colloquiallisms commonly used in English and Greek. If we included every single one of them, the hints and alternatives would be so full that learning would be very close to impossible. Sometimes, it's better to stick with the basic translations, not their alternatives.
(After discussion with a couple of native English speakers, there is no need for with credit card be included in the alternatives.)
Thank you for your encouraging words. Yes, we realized from the start that we had to be rather rigid is what we accepted as correct otherwise we'd be bogged down in an avalanche of regional, and colloquial expressions. We do of course accept a lot of alternatives but some just cross theline.