The problem is not that "racism" has no coherent meaning.

I thought the definition that someone got from Google elsewhere in the thread was fine. The only thing that definition leaves out is what people believe about the claim that "racism" labels. Some believe that it is true and some believe that it is false, the strength of their belief either way varying in proportion to their desire to exclude from discussion the question, "is this true or false?"

We might better ask, "Why do they respond so badly to this particular word?" I suspect the answer has a lot to do with fear of being accused of something vile.

Generally, they are being accused of a belief that their accuser thinks is vile, so vile that the very question of whether it is true is also vile, so vile that it must never be discussed, and it is quite clear without further explanation that that is what is meant.