I think most of the time we talk about curiosity it's how we see it's evidence in others which is not the desire itself, just purely the act of spending time and focus on something not 100% necessary.

Then there's the desires behind this action which can be hugely different and vary one day to the next. I think people who focus their curiosity on a single area tend to succeed outwardly for it. But people who have broad curiosity ('trait curiosity' as you put it) have enriched and possibly happier lives for this.