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I prefer calling it "floor-index" vs "ceil-index" rather than 0/1 or offset/count based

Both 0 and 1 indexing aim to answer the question: "what is the index of the 1st element?"

floor(1) = 1 and ceil(1) = 1 so I'm not sure how your names apply.

floor and ceil ideas apply to age and calendar years nicely because the underlying value, time, is on the real number line.