As I recall, the Labour Party constitution always spoke of workers 'by hand or brain'.
On that basis, you could be a Professor at Oxford and still be 'working class'. I don't recall any reference to employment status either. So I don't believe the self-employed were ruled out.
It's rather sad that this country is still obsessed by class in 2024. It's the root of many of our societal problems. It seems to spring from a very common desire to benchmark oneself against others. And, by the way, working-class snobbery exists too, and always has. The desire to look down on others is not restricted to any particular class.