and numerous other metrics for the BLS -- we'll see that report tomorrow morning
https://www.bls.gov/cps/
The Current Population Survey (CPS) is a monthly survey of households conducted by the Bureau of Census for the Bureau of Labor Statistics. It provides a comprehensive body of data on the labor force, employment, unemployment, persons not in the labor force, hours of work, earnings, and other demographic and labor force characteristics.
I don't know what the Census Bureau does in the BLS monthly jobs report beyond doing the survey; for all I know, the BLS then takes the raw survey results and compiles it all and seasonally adjusts it and so on, but anyhoo, thought I'd weigh in.
As for loss of employment related income, yes, that's no surprise, but at least according to this, it was more than made up for by stimulus and unemployment insurance spending for the 2nd quarter as a whole:
Personal Income increased by 7.3% in Q2, while personal income without government support decreased by 6.1% (graph), 8/6/20
https://finance.yahoo.com/news/stimulus-checks-coronavirus-personal-finance-expert-advice-201536967.html
The Bureau of Economic Analysis produces the Personal Income figures
This is their latest news release. It also has a graph of the month by month changes
https://www.bea.gov/data/income-saving/personal-income

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I have no explanation for the widely differing number of people claiming unemployment benefits for the 2nd week or more, compared to the BLS's unemployment in millions or unemployment rate in percent.
From Richter in the OP linked article
There is a huge difference between the 30.2 million people on unemployment insurance rolls, as per the Labor Departments weekly tally of unemployment claims, and the assertion that only 17.8 million people are unemployed, as per the Bureau of Labor Statistics.
His article is dated 8/3/20. Actually it's gotten worse, according to this morning's unemployment insurance claims report:
https://www.dol.gov/ui/data.pdf
The total number of people claiming benefits in all programs for the week ending July 18 was 32,118,678, an increase of 1,302,816 from the previous week. There were 1,707,267 persons claiming benefits in all programs in the comparable week in 2019.
Thread on this morning's unemployment insurance claims report:
https://www.democraticunderground.com/10142554593