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You could just turn this entire argument on its head.
The people in the Pittsburgh synagogue were not targeted for being Jews but for being American Jews. HIAS is not an Israeli institution. The Pittsburgh shooter didn’t have a problem with Jews per se. He had a problem with Jews in the US because they vote Democrat and support mass migration into the US. There is no evidence he had any problem with Israeli Jews. Most Israeli-Americans vote Republican. Israel will soon to be home to most of the world’s Jews. Israel’s Jews lean right. UK Jews and French Jews lean right. American Jews are targeted because of the policies they support, not because they are Jews. That was the Pittsburgh killer’s motivation. He explicitly cited Jews pushing left-wing policies. White supremacists don’t shoot Jews in synagogues in Israel or even in white countries like the UK or France.
Likewise many “right-wing antisemites” just had a problem with Jews pushing left-wing policies in their countries and being overrepresented in leftist movements like communism. If Jews just get out and move to their own country, have friendly relations with European nationalists, and build a wall to keep out third-world migration then they no longer have a problem. They are not actually antisemitic.
All jokes aside, I think Pat Buchanan was an example of someone who was in some ways antisemitic, but really not on a personal level. He didn’t really hate Jews but he was just annoyed with American Jews being overrepresented among Democrats, secularists, and war hawks. This video is good.
I also think the question of what is and is not “antisemitic” is not a question with a clear answer. I’m sympathetic to the argument that words like antisemitism, racism, etc should be used sparingly. But yes this is an antisemitic terrorist attack.