Hi John- Good question. It is certainly true that if you go far enough back in history both the Jews and the Arabs descend from Semitic tribes that migrated into the Middle east. I am not sure why the Jewish people, exclusively, started being referred to as Semites (I obviously need to do some research over the next few days).
When the Hebrew tribes, many of whom lived in Judah (today Judea). embraced monotheism and began to worship Yahweh (YHWH) a clear separation emerged between them and all the other Semitic tribes who still practiced polytheism. Of course the Arab Semitic peoples of Saudi Arabia embraced monotheism when Muhammed and his followers introduced Islam.
I think the current mainstream media may be unaware of all the historical particulars and I think they are continuing to use the nomenclature that has been in place at least since shortly after World War II. I am not sure why the Arabic peoples never adopted the term Semite for their own use. I do know that the Nazis referred to the Jews as Semites, but I am not sure when that term was first introduced or by whom.
this is a tangential issue , but it is worth mentioning because it adds to the confusion. Catholicism is a religion whose adherents are of many different races and nationalities. Being Jewish is not so simple. There are Israeli Jews (as well as non-Israeli) (Semites by blood) who are secular - professing no faith and who do not observe the tenets of Judaism.
On the other hand, it is possible for people of any racial or national background to study, practice, and be accepted into a Jewish synagogue and practice the Jewish faith, in other words convert to Judaism. But are they then considered Semites? More importantly, do Arabs consider them Semites?
So what does anti-Semitism actually mean? What are we to make of this confusing and overlapping set of terms? I am sorry I don't have a simple clear cut answer; it is a complicated and confusing situation. But thnak you for an interesting and important question.