Write a function boolean_xor(a: bool, b: bool) -> bool that takes two boolean inputs, a and b. The function should return the result of the exclusive OR (XOR) operation on these inputs. In other words, the function should return True if exactly one of the inputs is True (but not both) and False otherwise.
For example, if the inputs are True and False, the output should be True because only one of the inputs is True.