Your question isn't really a history question — it's about the present-day distribution of people, so /r/AskAnthropology is probably more apt.

But let me just introduce a little bit of historical insight here. There are two issues I have with this map: 1. the definition of whiteness, 2. the decisions made about visualizing it.

On "whiteness": racial categorizations are always difficult, because race is not (despite much belief to the contrary) an actual biological or natural category. The anthropologists of 100 years ago, for example, did not recognize any singular "white" race, but instead tended to break European "races" down more finely. For example, William Z. Ripley's Races of Europe (1899) suggested that there were three main European races (the Alpine, Mediterranean, and Teutonic), whereas others in his day (like the Frenchman Joseph Deniker) thought there were dozens of European races. The notion of "whiteness" as a unifying character emerged over time, and became prominent in the United States only in the interwar period, as a way to solidify a sense of racial-economic security against "outsiders" (African and Asian Americans, among others).

So we should be careful in such categorizations today, and ask why we are trying to make them, knowing that they are going to be somewhat arbitrary and that the history of such things is fraught. Sometimes there are good reasons to invoke the cultural-economic-biological-geographic pastiche we call "race" — just because it is not "natural" does not mean it does not have meaning (money is not "natural," yet it shapes our world), or does not mean it cannot be useful (using racial categories for rapid identification can be very useful, for example, as can using them for identifying historical wrongs). One can see, as an aside, the difficulty the category presents for the author in their discussions of their methodology, and the subsequent arguments about it in the comments section. The US Census racial categories are particularly historically tricky, and make for tricky modern-day assignments (e.g. with "Hispanic" as an ethnic group that can be coupled with "racial" groups). Again, the Census categories are trying to address specific questions relevant to society — that doesn't mean they're bad, but we should remember they are not in any way transhistorical categories.

On the visualization, I would just note that this kind of map, where you shade large political boundaries by percentages, can tend to obscure more than it reveals. It is of historical interest that some of the first methodologies for cartograms (data-maps) evolved out of questions about how to represent race (the aforementioned Ripley was very important in these matters, and wrote on them at length in his day. Personally I am not a huge fan of this approach, because it reifies political boundaries that may not have a lot of population-based meaning (as the author themselves seems to recognize at times), and can homogenize the data. One might compare the Racial Dot Map approach, which seeks to show more granular data. The RDM uses the same kind of data as the author, but instead of just filling things in with a broad color for percentage it uses the relative percentages as an algorithm for a dot fill. The RDM has its own issues (because it is reliant on US Census data, it has "compressed" some of the racial categories in odd ways), but it serves as an interesting contrast. One gets a very different sense of the "whiteness" of the United States from such a visualization, as compared to the blog author's map (at a bare minimum, one sees how sparsely populated most of the "white" part of the country is).

I think, as an aside, you have misread the author on some points. The author does consider European-origin South Africans to be white — they say only that the percentage is actually quite low on a boundary-by-boundary basis. That seems true, but is also a reflection on the difficulty of picking a good visualization methodology. There is no one perfect way to visualize any piece of data; I teach courses on data visualization and one key point I always emphasize is that you are always making certain choices when you do a visualization, and very slight (and intellectually-justifiable) tweaks can make the data look completely different. The question to ask is: What's the goal? What's the reason? What's to be gained in doing it one way versus another?

On the evolution of racial categories in the US, I am fond of Matthew Pratt Guterl's The Color of Race in America. On Ripley, Deniker, and the use of race in cartograms, alas, that comes from unpublished work I did years ago in graduate school... maybe someday I will clean it up and get it published, because it is quite interesting stuff!