While I do not know if Icici mine works for normal Icici Banks offers (I don't have mine cc; but Ig it should work for normal Cashback offers; As far as I know only Amazon Pay CC does not work for Icici normal offers )
Coming to their plans, I think you should get the extra cashback (unless otherwise specified) but do keep in mind tht extra cashback has capping of ₹200/₹750 depending on your plan.