"Show, Don't Tell" doesn't mean "use pictures instead of words," just for the record. You can "show" using dialogue. "Show, Don't Tell" just means to let the reader see evidence of something and draw their own conclusions rather than telling them how to feel.
It's not always applicable -- as a silly example, having a character say "My name is Jack" is a bit of exposition, but if you try to figure out how to "show" that instead of directly saying it, you're tying yourself in knots for something that's really not worth it.
So, for the true name vs. nickname thing, I don't think it'd be a sin to have a note, as long as it's efficient and doesn't insult the reader -- like, a protagonist is introduced and a caption says "his truename is Sparrowhawk, but since truenames are sacred, he's called Ged," then that's pretty efficient and lets the reader come to some conclusions on their own -- now, a character calls him "Sparrowhawk" and we immediately understand something about that interaction, and have to piece together on our own what kinds of people are allowed to know or use that name. It still allows you to "show" the significance of that practice.
On the other hand, if your footnote is also a short explanation of this practice and its cultural implications, I think you'd be over-explaining, and explaining those things wouldn't have the same impact as watching interactions and piecing together exactly how the different names are used.
As DMBrigman and shiftingverses pointed out, it's usually possible to fit this sort of thing in the narrative -- if you just use the different names, I think folks would miss the significance of the Truename and assume it's just a nickname, but if there is space in the narrative for, say, a character to ask about another character's truename ("I can't tell you my truename, I just met you!" is exposition, but isn't weird for characters who know about truenames to say to each other), that can make the whole thing feel more natural.
In general, I think anything you can let the audience put together in that way feels a lot better, and makes your world feel real in a way that constant footnotes do not. Bringing together characters who feel different ways about established pieces of the world makes it much easier to talk about and also puts those pieces into context. But it's also sometimes not worth tying yourself in knots trying to avoid saying a single sentence directly -- so I think what's the best way to convey it is going to vary from example to example!
It's not so much about restraining yourself to "the drawings and dialogue" specifically, and more about looking for places for this cultural or world detail to be demonstrated through interactions and through the story itself!