We've had this discussion lots of time on other threads but, IMO, not providing complete information to a boat crew can be a bad practice. The crew needs to plan their resources and make decisions that depend on evaluating the divers on board based on the information they have.
I suppose, if they are stupid enough to just ask for "a card" then all you have to do is show them "a card"...any card. But, for all but the most basic of dives (like a pool maybe) I'd be concerned about diving with a cew who is that clueless.
Assuming that when they ask for a card they are really asking what they should be asking, I try to provide the information that they should be asking for.
Aside from that, cards are for gaining access so when I travel, I carry the cards that will get me the most access and I wouldn't even know where to look for my AOW or rescue card.
I know that some pros and experience divers worry about being put to work but I'm not likely to accept any offers of employment on my vacation time unless the money is really great. Since I've never seen "really great" money offered in the recreational dive industry, I don't worry too much about that and I don't dive with anybody that I don't want to dive with.