It's simple. If you are referring to scarce and rare as rarity scale descriptors, it means whatever the author of the scale intended. If you are using the words to describe a coin generally, rarity is the amount extant (lower amount increases rarity). Scarcity is the amount extant relative to collector demand (higher demand increases scarcity). You can have a coin that is scarce but not rare (plentiful but highly sought out by collectors), a coin that is rare but not scarce (very few exist but nobody collects), a coin that is both scarce and rare (very few and highly prized), and a coin that is neither (cheap common coin).
