Of course we all love Nell Irvin Painter, because there would have to be something fairly wrong with you as a person not to. That said, there are some alternative answers to this question of "whiteness", one of which can be found in the headings of census forms over the decades.
I'll use Family Search for this, so you don't have to pay money to see them.
Here are the headings for 1790 to 1860:
And here are the headings for 1860-1930:
And finally, the most recent census form which is publicly available:
I'm sure if you look around, you can find blank copies of later census; I'm sticking to these because of the time frame that Nell Irvin Painter looks at in her historical overview. She says, for example: "In the mid- to late-19th century, the existence of several white races was widely assumed: notably, the superior Saxons and the inferior Celts. Each race — and they were called races — had its characteristic racial temperament."
While Nell Irvin Painter is ABSOLUTELY CORRECT, there were other definitions of whiteness in use over the same time frame, and those definitions are on display in the census categories.
Let's take a closer look.
In 1790, there are "free white males" over and under 16, free white females (all), "All other persons" and, as a separate category, "slaves". I'm not 100% sure, but I am reasonably certain that "all other persons" was not intended to include "slaves" or free white persons of any age or gender. I'm betting Native Americans, Chinese, and presumable free colored persons were all lumped into this category. (But if you know otherwise, let me know in a comment and I'll edit this post.)
The same headings were used for the next two censuses, 1800 and 1810. Now free white males are broken into under 10, 10-16, 16-26, 26-45, 45 and older, same with free white women. Again, everyone who isn't a free white or a slave, and finally, slaves. I'm not sure what was going on with that fine breakdown. 45 and older likely constituted "too old to draft"/too old to reproduce. Under 10 is too young to work, and the remainder give a good demographic view into how many people you can expect will be producing babies/available to draft. But someone may have more information about what they were doing with ages.
In 1820, the free white gender divided with age subdivisions continue, but we have other categories thereafter. "Foreigners not naturalized", "agriculture", "commerce", "manufacturers", "free colored" and finally, "slaves". Your guess for a lot of those may be better than mine. Immigration is starting to be an issue, but not enough of one to yet collect birthplace/ethnic group. The industrial revolution is taking off. "Free colored" has become a large enough -- and contentious enough -- group that it is important to count. This was the era of compromises on what happened when black people weren't slaves in the south: could you kidnap them and bring them "back"? Did their slave status persist when transiting through free states with an owner?
In the 1830/1840 censuses, the age breakdowns have been "rationalized" to 5 year intervals. Free white males and females broken up in 5 year categories to age 20, and then decadal intervals to 100, then "over 100". It's important to note that up through the 1840 census ONLY THE HEAD OF HOUSEHOLD was named in the census. Other members of the household are only tick marks in these various columns. After that ludicrously long breakdown are three remaining categories: slaves, free colored and foreigners not naturalized. (If you are asking questions about how the census was counting Native Americans or Chinese immigrants or their biracial/multiracial offspring in these decades, I have nothing for you. Believe me, I'm gonna be treating this post as a source of research questions.)
In the 1850 census, we finally get names beyond the head of household. This particular resource for census headings is unfortunate, because I'm reasonably certain that the Slave Schedules had other information on them. Now that we have names, the lines are much simpler: age, sex, color, occupation, finally! birthplace, marital status, school status, literacy
The 1860 census is similar -- the only difference is that in 1850 there was a box for value of real estate and in this one there is that plus a box for value of personal property.
From basically the first national census until the dawn of the Civil War, the census was counting you as white or slave, and, almost as a footnote, they were counting other people. They were counting white or slave, including some age information, basically to determine representation at the federal legislature (yeah, want to be offended by something, that 3/5s thing is worth being offended at). As immigration grew and the Industrial Revolution slowly got started, more data was collected, with birthplace becoming of interest at the middle of the century.
What happened after the Civil War? In 1870, there are familiar boxes (name, age, sex, color, occupation, value real estate, value personal property), birthplace, but now there are also: father foreign born, mother foreign born, month born in census year, school in census year, literacy, "eligible to vote". (Bunch of questions right there, amirite?).
In 1880, name, age, sex, color, and some new categories! Where before, there was a question of married or not, now they want a breakdown. Western divorces are easy to come by and the guv'mint wants some data! Single, married, widowed, divorced. Relationship to head of household (I love this field). Occupation, literacy, now they want the place of birth for both parents. I would consider 1880 to be the beginning of ethnic tracking and identification in the census. Before 1880, it was all about whether you were foreign born or either of your parents were foreign born. But in this one, they want to know _where each parent came from_. You can argue that as soon as they were collecting birthplace, ethnicity was A Thing, which moves the beginning of ethnic identification and tracking to 1850. But in 1850, you were only "ethnic" if _you_ were born in another country. In 1880, you are "ethnic" if either of your parents was born in another country.
1890 makes us cry. We shall draw a veil over 1890.
Beginning in 1900, they start to collect information relating to birth rate: number of children born to a mother, number of them still living. Marital status is back to a code in a single box, rather than several tick boxes, but they want to know how long a couple has been married. Information is being collected about Veterans of the Civil War -- there was legislation being debated about how who should be eligible and for what. For the first time, there is a question about whether the person can speak English, and literacy has become more complex: instead of "cannot read or write", it is now multiple boxes, one each for read, write. There are also boxes for unemployment and students. Citizenship status is broken down: immigration year, number of years in the US and whether naturalized.
It was possible to collect a lot more information in the census, first with the drop in the price of paper in the first half of the 19th century, and then with automation of counting using Hollerith cards in 1890 (*cries*). So some of the additional data collected can be attributed to, "hey, we _can_ collect this, and we sure want to". But I would argue that even that supports a thesis that how people are treated, identified and grouped evolves in a larger context. First, the census existed to determine representation in the legislature. Functionally, if you were in the United States in the late 18th century, you _were_ considered a citizen and you _could_ vote, if you weren't a slave and you were a man. "Whiteness" was a broad category to which all manner of people could belong. In the leadup to the Civil War, as abolition took hold in some states but not others, and as generations of interracial children produced more generations of interracial children, the difficulty of drawing a bright line produced first new categories (many of which we have abandoned and were never reified in the census), then a war, and then finally a return to colored/not colored, with ethnic data collected on _everybody_ (all that birthplace, father's birthplace, mother's birthplace stuff got filled in for everybody after the Civil War).
Starting in 1910, immigrant communities in cities of sufficient size and complexity increasingly allowed children to grow up not learning English. Information was now collected on the language of each person. This is also the first census to explicitly collect information on disability (blind - deaf - mute).
In 1920, the ethnic information on parents is no longer just birthplace, but also on parents language -- this would be even for parents who never came anywhere near the US. This, then, is finally the full-flowering of ethnicity in the census.
The 1930 census gives an indication of the farm-to-city trend (a column newly devoted to "live on a farm") and new technology (radio?) But ethnicity information is declining, replaced instead with more detailed information about veterans ("which war"). And, disturbingly, an entire schedule for unemployed persons', and an entry to put the number in to connect them with the full census.
A brief overview of the headings of the US national census tells us many things about our history. The data we collected at the beginning was vestigial -- but it was ambitious for its day, intending to be a decadal event. It was driven by a need to calculate representation in a rapidly growing, new country. As paper got cheaper, and we automated more of the enumeration, we could ask more questions, and we did. First, we collected everyone's names, even the children. And we started to pay attention to birth rate, and to pay attention to the growing category of people who were neither "Free whites" nor "blacks", because this category of people was the locus of so much political contention that no political compromise lasted for long, and more than one national political party met its end attempting to avoid abolition.
Once we were through those bloody years, ethnicity became the next focus of attention, eventually crystallizing in a realization that the crux of the matter was shared language, and the solution was public education.
You really can _see_ how right Nell Irvin Painter (and just about anyone else with any sense at all) is, when she calls "white" the default category. And she's not wrong to point out that privilege for some people at the expense of others has Got to End.
Might be time for another name change, this time away from the word "white". The assimilationist project which is the melting pot can fix the privilege problem, but we may be getting hung up on a term that needs to go away.