the one that you can't apply from official website is always co-brand card (these are the cards either exclusive to that site or can apply both places).
But AFAIK, in Axis there is no difference between offers applicable to core cards and co-brand cards, that differentiation so far happening with ICICI only, and no other bank follow that approach to limit using their cards for specific offers.
Look for 'partnership cards' section.
Regarding c2c, can't really say even APay sometimes works and sometimes not, it depends on that time.