"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.)
Dear jonny12345, let me explain something to you. We are a team of Greek English speakers all of whom have a good deal of knowledge and experience to have been chosen by the Duolingo staff to create and maintain this course.
We all work on the set of guidelines given us by Duolingo. As you will see from reading some of the comments we're more than willing to add, edit and remove translations when we consider them necessary. Stubbornness is not one of the principles of a serious educational course.
We do however expect to be treated with the respect that our efforts deserve. Please read the Duo guidelines found at the bottom of the pages.