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Actually, no, the type of prop firms I am describing are typically small (in terms of personnel, not in terms of AUM) hedge funds which run some manager's own money and some outside money.
Facilities for day traders are a different thing entirely and I don't talk about them here.
I guess we are talking about two different things then, I have never heard of any one talk about the term prop firm in your sense. Doesn't make it wrong though.
Edit: If you are describing a small hedge fund that manages the managers money and outside money, then its just a hedge fund any way you cut it. Proprietary means only the firms capital is used, and for no clients.