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It's interesting how people try to redefine the word joke. If there's nothing funny about it, it's not a joke. So maybe they mean that they were trolling just for fun, which is not the same thing.
The second point is that if they say something horrible and then you call them an asshole and then they complain that you've gotten too sensitive, which is a very common chain of events, it actually shows that they've gotten too sensitive. If you're not supposed to care about what they said, why should they care about what you said? ... But of course actually they don't care about what you said, and they're just grandstanding. But even if we accept their internal logic just for the sake of argument, it still makes them giant assholes.
If there’s nothing funny about it, it’s not a joke
This doesn't excuse these people, but this is a bad way to define a joke. Funny is a subjective thing, a matter of taste, there is no objective "funny" to measure against to see if they qualify.
A lot of language is subjective, and jokes are notoriously subjective. They translate horribly across cultures and languages.
If you have a different definition in mind, I'm all ears.
Jokes are about intent. There is no way to define what does, and does not, qualify as a joke without enough error as to make is useless. It is all about context, subject, timing, etc.