Tbh, I think the term Middle East is so imprecise that it should not be used at all. No matter how you try to define it, you're going have to end up with a definition that either exclude countries dead smack in the middle or excluded for arbitrary reasons.
I do indeed think that in Europe even though east Asians are the first thing people think of when they hear Asian, but that the vast majority will include south Asians as well.