You are right, there are Jews who aren't Semites an there are Semites who aren't Jews. The term “anti-Semitism” came up near the end of the 19th century [1] when it was all the rage to find scientific-sounding words for whatever you do or sell. “Anti-Semitism” sells better than plain “hatred of Jews”. I for one wouldn't give racists the satisfaction of using their own choice for cloaking their prejudice and stupidity in pseudo-science but the term seems to have taken hold everywhere.

[1]: https://www.britannica.com/topic/anti-Semitism

Languages are Semitic, people are not. These categories are outdated to the extreme.

Sometimes words don't mean exactly what their etymology suggests. So I'm sorry but, objecting to using the world 'antisemitism' to refer to "anti-Jewish racism" comes across as trying to deny Jews the ability to describe their own oppression

Hanlon's razor applies: "Never attribute to malice that which is adequately explained by stupidity." Many or most people incorrectly assume that anti-Semitism has something to do with Semitic people because the words (but not their meanings) are similar.

Hard disagree. One of these (anti-Semitism) is far more common than the other (references to Arabs as "Semitic"). It's like saying that you've only heard of "negro" as a color. In a culture and a place (the United States) where that is absurd, assumptions of malintent and bad faith are a better heuristic.

I don't mean to imply that the word makes any sense in this context. However, if one chooses to complain about Israeli abuses of Palestinians, it is a fact that one will be called "antisemitic".

Isn't "Caucasian" a specific ethnicity of the region between eastern Europe and western Asia?

The point you bring up is an example of "fallacy of equivocation".

Yes, words are polysemic. In this case, 'anti-semetic' is seen as "anti-jewish". That's one sense of 'anti-semetic'. You can also add another sense of that word: "against speakers of Semetic language or people". This sense is not used when people use 'anti-semetic'.

In Europe, there has been centuries of discussion about jews: it was called "the Jewish question". Anyway, this anti-semitism started as a product of religious rivalry (which religion is True? Christianity or Judaism). Then extra features were added, etc.

Yes, they are also a Semitic group, but "Anti-Semitic" has come to refer to hatred specifically against Israelis.

Maybe because that hatred is more widespread than hatred against all Semitic people...?

Actually, "anti-semitic" is against "jewish" person (as a religion) and not against "israeli" citizen (as a nationality). The last one is called "anti-sionist"...

The fact that Israel is a nation "home for jews" doesn't help, as some people think that every critic against Israel politic is the mark of "anti-semitism" even if religion has nothing to do with. Sometimes it might be the indicator of "anti-sionism" and sometimes... just the indicator of political difference! After all, you can criticize any country for its politic without critizing the counrty "per se"

Moreover, a lot of people in Israel are not jewish but muslims.