While everything you say is true, Lewis traces the most virulent anti-semitism to it's roots in western Europe. It's important to note that he defines anti-semitism very carefully and rigorously; basically, he says that the natural rivalry of Islam and Judaism occupying the same territory and vying for political power and social dominance is not anti-semitic. In his view, one can be anti-Zionist, or opposed to the state of Israel or it's actions without being necessarily anti-semitic. He defines anti-semitism as the deliberate mythologizing of the Jews as evil incarnate-- the sorts of blood libels found in a book called "The Elders of Zion" and other such tracts, which allege that Jews drink the blood of non-Jews, and use it to make their religions foods-- other crazy stuff like that. Basically, he sees anti-semitism as the belief that Jews must die because they are inherently evil.
I think he takes a very reasonable tack, actually bending over backwards in the effort not to label anti-semitic any act which could possibly be explained by another motive. He gives the example of the murder of a group of Israeli tourists (men, women and children) by an Egyptian policeman (I can't remember when-- maybe the '70's or '80's?) He doesn't call that an anti-semitic act, explaining that any society can produce a lone nut. However, the Egyptian press (government-controlled) did NOT give the facts of the incident accurately, and the perpetrator became a huge folk hero in the Arab world. Lewis does see that, quite rightly in my opinion, as anti-semitism. The current myth popular in the Arab world, that the 9/11 attacks were the work of the Mossad, is also anti-semitic.
I'm not explaining any of this as well as Lewis does-- I would highly recommend reading his book.
You explained it very well.