Won't that be fairly common? Java's hashCode and the .Net GetHashCode both return 4 byte ints and there are presumably a lot of HashMap and Dictionary objects out there using short strings as keys.

It's very common; the norm even. My point was that "A hash (function) is any function that maps a big variable-length data structure to a smaller (fixed-length) data structure." is incorrect; the source size isn't relevant.

The source size is extremely relevant, in the property that the function is able to work with variable sized input including large input. This definition has obviously embedded its purpose unto itself. The reason we have such cryptographically secure hash functions corresponds to the need to digest large data.