You made me curious so I did some very quick research. It seems that contestants are considered independent contractors in many, many cases. There may not be much distinction, and when there is a distinction I don't know what that would always be, and it probably varies by state. Reality show contestants appear to generally be considered to be independent contractors, pageant contestants appear to generally be considered to be independent contractors, a marathon runner was considered an independent contractor, and various other contestants were as well, and all or most of them were competing for a prize, just like in pool. I did not find any specific mention of pool.
I also found that the difference between an employee and an independent contractor is actually very muddy waters and not well defined at all by law, and it also varies by state, even though it is not well defined and is muddy waters regardless of state. There is certainly more to it and not quite as clear cut as what you laid out, like whether your boss controls or directs your movements during the day, and many, many other factors, again depending on the state.
The one thing that is certain that you mentioned, is that what pool players playing in a tournament are officially considered to be has exactly zero to do with what was being discussed in the thread which was what behavior a promoter should accept or be expected to accept from them. Regardless of what they are considered, it doesn't change a thing.