Racial theories of the late-19th and early-20th centuries tended to be "divide up the whites" approaches to race. They were often written by Western Europeans (British, French, Germans) or by Americans who were in the Northeast USA where there were heavy influxes of immigrants from Europe around this time. They strongly reflected national and class prejudices of the day, and frequently divided up the European "races" into categories that put people from Southern and Eastern Europe into dramatically different categories than those of Western and Northern Europe. The canonical example of this is the American lawyer/ecologist Madison Grant, who was himself drawing on earlier work, and whose work was taken up by the German Nazi theorists as well. So under these rubrics, Italians, Poles, Slavs, and, of course, Jews, are "non-white" to the same degree that Scandivians, French, and British people were "white." (Germans were often a half-way point whose inflection depending on who you were and what the exact time period was.) These categorizations were not idle ones; they came with consequences. Grant's argument was that all achievements of the world had been caused by the Nordics (northern "whites") and that the USA would be destroyed if it did not immediately invoke a blockage of immigrants from these "undesirable" parts of the world (which the Immigration Restriction Act of 1924 did).

All of this, it should be noted, was a form of scientific racism and not rooted in actual biology. At the time, there were certainly those who said so, and alternative racial theories as well.

Anyway, the big shift in the USA to a bipolar, "white/black" racial theory (or, if made more global, a "five color" racial theory), happened in the interwar period between WWI and WWII. This is seen as a response largely to the Great Migration, wherein many millions of African-Americans moved out of the South and into the cities of the West and the North. This radically realigned the class values, and there became a premium in a solidified "whiteness" as a racial/class/tribal identity. Under this model, Italians and Irish were now "white." Again, these were not idle theories: they explained to those who believed in them why "whites" should secure their economic placement at the expense of others, why certain politicians should be supported over others, why certain types of violence should be tolerated and encouraged, and why policies like voter disenfranchisement, segregation, and anti-miscegenation were both right and important.

There is much more that can be said about the way in which these kinds of identity categories have been deployed, invoked, attacked, etc., over the decades, and the American context is not the only context for this (though it is a major one, and one that was exported to many other places), but I want to give you a flavor here of the ways in which these kinds of racial taxonomies have never been "natural" and they have never been self-evident. They were constructed at various times, by various theorists and writers, to deal with the contexts they found themselves in, often as a justification for policies they desired. Racial categories are not biological categories — genetically, humanity is a big blur from one appearance to the next, and anyone deciding that there is some kind of line that makes one specimen a "pure type" and one an "admixture" is being arbitrary, and usually for some kind of political reason. Beware falling into the trap that "white" is some kind of real biological category! It is not. It is a historically, socially, and economically constructed category that has immense power, as I am sure you have experienced. (Construction does not mean "fake": money is a totally socially-constructed entity, one that loses all value in absence of a society, and yet it also has immense power.) But do not confuse any of these categories with being pure biology.

For a very nice discussion of this history and changing views of race, see Matthew Press Guterl's The Color of Race in America.