Races are like Russian dolls - there are nested races inside one another. In one context, it makes sense to talk of the 'Israelite race' just as it makes sense to talk of the 'Anglo-Saxon race' or the 'Iberian race'. However, the term 'the White race' is typically taken to refer to a larger ethnic set that includes such sub-races as the Anglo-Saxon and Iberian races.

My argument is that none of the evidence that you've adduced gives us any reason to think that whatever was called the 'Israelite race' is unlike the so-called 'Anglo-Saxon' or 'Iberian' races in that it forms a subset of the White race. Moreover, a great deal of evidence - namely, the facts on immigration from the early 19th century, which included both Sephardi and Ashkenazi Jews; explicit discussions of whether or not Hebrews counted as part of the 'Indo-Aryan' race from the early 20th century; and British ethnographic studies from the 18th century - all lead us to believe that whatever the Israelite race was believed to be, it was believed to be part of the larger White race. This suggests that it was believed to be a race on the level of the Anglo-Saxon or Iberian races, and not one on the level of the White, Black, or East Asian race.

In any event, this sort of sociological history about what (Ashkenazi) Jews were considered to be doesn't settle the question of whether or not those considerations were correct - it may turn out that Jews were erroneously thought to be non-white, or erroneously thought to be white, and so on. The question what race Jews actually are is a biological one.