Your comment is the first I've seen actually providing intuition about what is happening. It doesn't help perhaps that the name itself is misleading as heck to computer people like me: it's not a filter as in stream processing or SQL.
- simple to predict / understand what it would do: Easily explained with pictures and hardly no math
- simple to understand at a higher level / see why it works: Not easily explained without math and fraught with bayesian vs optimization vs EE-type approaches
- simple to understand well enough to use: Not easily done without other relevant math that is not covered in KF explanation, e.g., controls, matrix analysis, Jacobians, etc
- simple to understand why the equations are named what they are, and why they work: Not easily explained without math and historical context that takes a page or so to explain.
> it's not a filter as in stream processing
As you say "filter" means 'remove noise', but it also means 'process in order of arrival', so it's similar to your def of filter.
So, we really need 4 guides.
filter: use measurements up to time t to estimate the state at time t
smooth: use past and future measurements to estimate the state at time t
predict: use measurements up to time t to estimate the state at time t+n