Earlier quoted context omitted.
To be honest I do not think that these word games are helpful at all. Throughout all of my mathematical education what has always helped me to keep my "mental workspace clean", was to never abandon the model. > and correctly taxonomising those propositions The correct taxonomy for a proposition is true/false and proven/unproven. I can not even fathom a mathematical model where distinguishing a "law" from a "fact" is…
I dislike the linked site. A lot. But counterpoint: Zermelo-Frankel with or without Axiom of Choice is a fair mathematical analogue to distinguishing laws and facts, in my opinion. Put another way, decidability is a large area of mathematical research.
What does ZF(C) have to do with decidability? Decidability is a question in any sufficiently complex system (Gödel's first theorem). And exactly this distinction is what I made for the taxonomy of propositions, you can group them into true and false and also into provable and unprovable. What would be a fact and what would be a law?
Regardless of that, in neither case the empiricism the site uses to define a fact would play any role.