Assuming you are running Windows 10, does something like the following show up in Device Manager?
>Memory technology devices
Realtek PCIE CardReader
If you don't know Device Manager then you could try checking Settings-Apps which should have Realtek Card Reader.
If not found, then you need to do something like get it from Intel's Download Center and install.
Once you have that, when you say you think you inserted it correctly, did you have the card label facing up - pins down? Did you press until there is a sort of click and almost nothing sticking out (maybe half-mm)?
When card is properly inserted, to remove you have to press in again (I have to do it with my fingernail) and the card will sort of spring out slightly (maybe 2.5mm).
P.S. I am not sure whether there is any default/generic support in Windows if you don't have that Realtek software installed. In various old hardware I would sometimes find that some software didn't really do much other than displaying more user-friendly icons for the card. My guess is that the software is important to install for this NUC, and that is why it is the first thing I discuss above.
I inserted the card as you suggested -- I don't hear a click, but it goes with with maybe 1/2 mm sticking out. Card is not recognized. However, with the computer on, when I push the card in further, it is recognized, but when I let it go, it is not recognized.
Could this be a problem with the card or the drive?
I'm not perfectly clear whether you are saying that you are sort of "practicing" with computer off. If so, that might make sense but it is just practice. What you need to make happen is to push it in (I have to use my fingernail) so that it sort of locks with that tiny amount (maybe half-mm) showing. Afterward when done, you press it again, it will sort of spring out (maybe around 2.5mm) - that is the release so you can then finish removing with your fingers. If you can't get those two steps happening, a most likely reason is you are using something like the flesh portion of your finger which isn't quite hard enough to reach that lock (or unlock) point.
The same technique is used when computer is on. If card is not "sticking" in after you move your finger away, then you probably have not pressed hard enough. It won't be recognized in Windows until it is sticking in (and you have the software driver as I described).
Of course there is always chance for defective NUC or card, but my guess is that you aren't reaching those click points. It isn't necessarily easy to deal with little cards with big hands.