The US Department of Health and Human Services also provides a website about bullying, StopBullying.gov. Part of the definition of bullying is that it "involves a real or perceived power imbalance".

This may also occur in the workplace. CDC offers a webpage on this. Here is a website which agrees with the OP that certain advice "like simply not caring" is not helpful:
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... advise the target to "get over it," "deal with it," "get a tougher skin," or offer any number of other equally useless and condescending remarks. - See more at: http://workplacebullyinghelp.com/stages-of-workplace-bullying.html
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For example, bullies will mock some feature or perceived flaw like someone's weight, height, accent, stutter or mannerisms. Imitating someone, or making a disparaging remark occasionally with the target in on the joke and laughing at themselves is one thing—repeatedly doing it after it is obvious that it is hurtful is quite another and is one of the tactics employed by bullies. Then they always have the cover of saying of the target, "they can't take a joke," or "they are too sensitive." In this way the target is slowly but surely made out to be a humorless, touchy, malcontented weirdo. - See more at: http://workplacebullyinghelp.com/mobbing-its-course-over-time.html

When advocacy goes awry, students may perceive retaliation and parents may similarly feel bullied by school officials, concerned that their children may be pushed out of school. A type of co-dependency may develop as families try desperately to make a poor relationship work.

There are many facets to dealing with bullying and negative emotions, depending upon the context in which it is occurring, and what the precipitating events may have been. Healing is important.