Floyd's cycle finding algorithm on SHA256
We usually use raw bytes, because this doesn't require extra step (convert hex to string). But technically speaking any representation will allow you to find a cycle (just a different one).

When talking about shortened-hash cycles, same rules apply: we usually use raw bytes just cut to desired length or we fill rest of hash with zeros, because that requires least steps to complete. But you can use anything that will cut entropy to desired value.