Raphael Lemkin did not fight for the legal term "genocide" to provide a "unique claim", & his initial definition was whittled down by other states in part under the claim that the Nuremberg verdicts (done *without* the term genocide) were sufficient. Other nations in the 40s tightened its range out of concern that too broad a definition would see them culpable for similar.
I recommend Lemkin's biography & 1940s articles for a clearer understanding of the term & its history.