Ah, I get it. We typically use it since it's the name of the camp, but I know a lot of us don't really call ourselves that, we just stick with demigods. Mostly for the same reason as your friend. If they arrive, I'll be sure to be careful to not insult your friend.
I mean we have some adults like Chiron and Mr. D who runs the camp.
Mostly, though kid doesn't feel quite accurate since I'm 17, a lot of us show up at camp around 12 or younger. We attract monsters easily so if we're out in the real world we have a high chance of getting attacked. A lot of us don't even make it past twenty, but it's not impossible.