While some oversensitive people can see bullying everywhere, there can be general consensus on that consists bullying.
Its desire to humiliate someone + desire to control said person or damage him psychologically.
You seem to be arguing that bullying requires an element of
sadism and while that might be the general consensus, we all know the general consensus can be, and often is, way off. Look at the general consensus on incels and the black pill.
But let me give you a hypothetical scenario.
Let's say you have an office worker named Joe. Joe is hoping that when his boss retires, he will get promoted and take his boss's place. However he has 5 coworkers who are also hoping to get promoted. He's working his hardest to be the best, but so is everyone else. So Joe instead decides he will do everything in his power to ruin everyone else's chance so he gets the promotion. He sabotages work projects of his coworkers. He spreads harmful rumors about them. He frames one of them for a horrible crime and ends up getting him convicted and imprisoned. However, Joe does not have a desire to "humiliate" them per se. He knows he's inflicting pain and suffering on his peers, and while it doesn't bother him, he also doesn't enjoy it. He is motivated purely by greed, not sadism.
Is Joe a bully? Y/N?
Personally "bully" wouldn't be the first descriptor I'd assign to him, but at the same time, I'm pretty sure that when you get down to brass tacks, Joe's behavior is downstream from the same genes that motivate the stereotypical high school bully. And your OP post is about genetics.
Anyway, if your definition explicitly requires a sadistic desire to humiliate, then by logical extension, those who inflict pain and suffering onto others while being solely motivated by greed, ideology, or politics wouldn't be "bullying". Maybe it's not by mainstream definition but once again, your argument is about genetics and I'd argue the same genes are responsible. No, I don't have a source for that, it's all just theoretical and nobody really knows for sure.
But even if that weren't the case, there's also the fact that the amount of pure damage done by the people motivated by greed or ideology is often far greater than those motivated by sadism.