That was an interesting video. I've corrected a couple ambiguous portions of your post, those who want to google Kristin can.
So in the video she's referencing 125kHz RFID which is a little different to 13.56MHz NFC. It's possible to do a similar thing (I modified a tag a while back with an LED instead of an IC wafer) but you'd have to play around with it to see if you got consistent results with cards.
With the rings though, totally different story.
You get a little bit of cross talk due to the proximity of the two inlays already but if you were to try to add in something like this to the ring itself things get a little more complicated - you'd have to have a system to switch it on and off if it's in the ring. Or to switch the IC out for the LED, lets say, because running the LED as well as the wafer doesn't leave much power for the thing you actually want to do and makes it slightly more difficult to get a good read. It'd have to be done in such a way that it didn't ruin the tuning of the antenna so that you could still read the ring when you want to. It's difficult. And all this inside the body of the ring.
You could go the other way by wearing an extra, ring cover over the top of the real ring but now it's just getting really inconvenient. Ring cover has to be large enough to go over the top of real ring, ring cover is going to be chunky and feel really off.