Here is the IRS article that covers this, including your "one of 3 categories" explanation, and the use of 1099 is here, as well:

https://www.irs.gov/government-entities/worker-reclassification-section-530-relief

Here's a nice article for statutory employee and statutory non-employee (independent contractor):

https://www.symmetry.com/payroll-tax-insights/statutory-employees-vs.-statutory-non-employees

And here's an article that seems to explain your comment about "they pay their own half:"

https://www.thetaxadviser.com/issues/2010/dec/zimmerman-dec10.html

All of this relies on the facts and circumstances for the specific situation.