All terms referring to ideas are made up. We made them up; the animals sure didn't. Your technical point falls on its face there.
Hate speech for sure is a real thing. Is calling someone the N-word hate speech? Not always. Context matters - there are some corner cases where it is clearly not.
But mostly, calling someone the N-word or other racial slut is a hateful act.
This word has such a long and nasty history that it is extremely clear its meaning. There are very few corner cases e.g. Hate Speech exists.
Your complaint is less that it doesn't exist, and more that mods are seemingly expanding the meaning by making judgment calls on the intent of speech.
Using another example to try and explain it, if someone said, "Why are the Inuit so lazy! They just don't want to work." (Something I cannot imagine anyone saying, btw) Well. That is a negative and purjorative generalization of a racial/social group, and whatever -ist label you want to put on it, that too is hate speech. If you walk it back a bit and get a little more wordy with the language, where it may not be as explicit making that generalization, a mod may or may not decide to ding you for it. It then depends on their reading of your intent.
Some mods get it right (I would say most), but some get it wrong, and there are a few who use TOS to enforce their own biases. That hardly negates the existence of Hate Speech.
Forum moderation is hard, as I said in a previous post. You will not always agree with the mods. And indeed the mods do not always get it right, and sometimes there are bad mods. The key is to try to first understand the TOS and second mind the social norms of that forum (just like you would at work or at church).
Above all, the same basic rule applies in online forums as it does in work or church: Just try not to be a dick.