For what it is worth, i'm paid by an agency and they are adamant that all the pilots are self employed but fly for the same airline.
Not sure how the rules work in other countries, but that is not true in the UK any more.
The primary criteria is the number of revenue streams you have. If you have less than three sources of income in the year, you will be designated as employed by the agencies you work for, and therefore a part-time employee. The days of a single employer paying a self-employed have long gone.