What truly matters here? As others have pointed out, what we call that curving staircase doesn’t matter in end of the world terms. But it DOES matter in the real world, in that whatever term is used is used to communicate an idea to the person paying for the end product.
So let’s just assume for the moment that we come to consensus here and say “It’s a spiral staircase if the inner post supporting the stairs is vertical and straight, otherwise it’s a helical staircase.” What have we accomplished?
Confusion - that’s what we’ve accomplished. The paying customer takes one look, says it’s a “circular” staircase, why are you calling it something else? And if we don’t answer that question perfectly, the customer gets the impression that we think we’re better than them. If we answer poorly, going into background information (like recounting this topic) - especially in a pompous manner - we’ve now alienated the customer. And if we’ve done this sort of thing regarding other details, the customer may become so sick of us that we get fired!
In the last week, since this topic was first broached, I’ve actually talked about it with a few friends with varying backgrounds. With the exception of a professional mathematician (who was fascinated with the nuances), the universal reaction was: “They’re all circular staircases - some more fancy than others.”
So in the real world, I’m going to limit my terminology to “Circular” and, if the inner radius is sufficiently large, I might start saying “Curving” instead of “Circular”.