I understand the academic underpinnings of that idea, but that's not how normal people understand the term "white." The Bureau of the Census, for example, certainly appears the believe that "white" reflects a racial category: https://www.census.gov/topics/population/race/about.html.
We rarely get the chance to see that the idea of whiteness comes from a need of protecting an in-group from exploitation, while excusing and justifying the exploitation of outsiders to that group.
The census is an interesting example because for the first 70 years or so, starting in 1790 the US census had just 3 "race/ethnicity" categories. You could be "Free white male/Free white female", "All other free person" or "Slave". Which is part of the reason for creating the category of whiteness. There's also some really gross stuff right through the 1800s where the census was tracking people with various percentages of "black blood".
https://www.pewresearch.org/interactives/what-census-calls-u...
I suppose I'm just saying that the census, as part of the machinery used to implement and maintain white supremacy, is not necessarily the best authority on race.