I have not seen the credit card type. I don't get out much. Sounds like they would be the cats meow.
I have a pretty good assortment of sharpeners from ceramic rods, to japanese waterstones, to DMT bench stones with even a few old norton oil stones tossed into the mix.
Waterstones are a lot of hassle to prep and clean up after, but they give a great edge and insane polish. I mostly just use them for the kitchen and fine carving knives.
For a pocket system though, I find the DMT credit cards hard to beat. The most useful tools are the ones you have on you the majority of the time and these tend to be for me.
If we're talking axe sharpening, you do have to exercise some caution since there's no guard whatsoever. But since it's diamond and fairly flat, you're not going to be needing to use much pressure and the risk of cutting yourself is minimal:
The whole kit (325, 600, 1200) in the leather pouch I made for it:
And the pouch next to my wallet (note that I scuffed up one side of the pouch so it can double as a strop):