"Bullying" is one of those grey areas for me.... It's not that I believe the behavior is "correct" or "appropriate" in any way. Rather the issue for me is how to prevent it or the consequences once it has been committed that falls into that grey area for me.
For young children, they do not have the life experiences yet in order to know how to deal with it, and at the same time the bully's words also tend to be more hurtful; the child doesn't fully understand themselves yet either, and so are more willing to accept stereotypes that the bully may use (ie "watching cartoons is for babies, and look at that baby wearing a Batman T-shirt" even though there are plenty of adults that enjoy cartoons or wear Batman T's). So for young children (elementary aged) I think it's ok for the school to step in and punish the bully.
However as we get older and begin to learn how to combat the bully with facts and logic, we have less need for an authority figure to step in. Sometimes a person may get their feelings hurt by what someone else says, yet it may not constitute a case of true bullying. Unfortunately many school districts classify "bullying" as any speech or action that hurts another student's feelings. A student may be charged with bullying by giving a pro-life speech in speech class, esp. if a girl in that class may be contemplating that very choice herself and gets uncomfortable or even hurt by the speaker's words. It is this over-broad definition of "bullying" that I have a problem with, esp. in later school years (esp. high school and even more so in college). We've come to a point where schools feel the need to cocoon their students away from anything that may hurt their feelings, even to the point of silencing legitimate discussion. Trigger-warnings, safe-zones, even calming rooms are now beginning to crop up because we're creating a generation that thinks it is their right not to feel hurt or uncomfortable by what another person says. It starts to deteriorate free speech, and strengthen those in authority who are able to deem what type of speech is hurtful and therefore "bullying" in nature.