A hate crime is basically one where the person being attacked is different from the attacker in some way, usually , although not always , in a way which is involuntary and that difference is the motivation or prime motivation for the attack. Not all differences are protected. Not all crimes where the parties are different are hate crimes.
Example 1. A fully able ,white, heterosexual Christian man and a black , disabled lesbian , Muslim, single mother live on the same common stair. He has frequent late and noisy parties- she has cats who pee on the stairs and children who drop litter. They can't stand the sight of each other. If they come to blows it is not a hate crime unless he uses insulting words about her race, religion or disability or indeed she tells him he's a dirty kaffir.
Example 2 A Glasgow Rangers supporter beats up a Glasgow Celtic supporter or vice versa. That is likely to be a hate crime because of the sectarian Protestant / Catholic rivalry. An Aberdeen supporter beats up a St Johnston supporter or vice versa - that is unlikely to be a hate crime as there is no sectarian rivalry between those teams.
Example 3 the 2 teenage girls who were convicted earlier this year of the brutal murder of a vulnerable alcoholic older woman after torturing her for hours did not commit a hate crime. Being a bit odd and being seen as a bit weird is not a protected characteristic.
Example 3 the people who bullied Fiona Pilkington to the point she killed her daughter and committed suicide did not commit a hate crime although the ones bullying her disabled daughter did.
Example 4 anyone regardless of their race, religion or sex attacking a white, Jewish middle class, able bodied man in the UK even a member of the House of Lords , because he is Jewish commits a hate crime.
Example 5 anyone regardless of their race, religion or sex beating up a Goth just for looking after bit funny is not committing a hate crime. Looking at bit odd and being seen as a bit weird is not a protected characteristic.