fit=crop and fit=cover only work well if the source dimensions are greater than 512×512. If either dimension of the source is smaller, the behavior is useless.

fit=cover will return a square (as requested), but will enlarge it (which you never want because it wastes bandwidth).

fit=crop will cut off excess pixels if either dimension exceeds 512, but it does not always return a square. (And contrary to the docs, it does not behave like fit=scale-down.)