Very much agree... but also want to be a little soft here. It's a very specific line of reasoning needed to discuss the difference between a "structured set", a

Very much agree... but also want to be a little soft here. It's a very specific line of reasoning needed to discuss the difference between a "structured set", an algebra, a "theory" (algebraic or geometric or otherwise), et al. Generally, we're all talking about the same sort of thing. But also, generally, there isn't one good formal apparatus for discussing the whole of these things without inducing just tons of complexity. Why? Because we want to talk about specific and complex things and we often want to use specific and complex language as opposed to working through 8 levels of encoding every time. So anyway, right there with you! But also happy to guide people down that path step-by-step.