We may have a definition problem between English and American usage.
I read (and answered the original question) with the assumption that "hate crime" is a crime commited specifically against someone because of their religion or ethnicity. Yes, because one group "hates" another, but I'd say that was based strictly on being taught ones prejudice and nothing to do with any real emotional basis of the type that occurs with jealousy or a crime of passion.
Semantics. I'm not happy with my definition using hate as part of the explanation.