Private (as in: not publicly traded) is indeed pretty much equivalent to publicly traded if it's a private ownership structure consisting of VC funds. But that's a startup bubble peculiarity, elsewhere private ownership either means being a subsidiary where the parent ownership structure transitively applies (strictly speaking, the VC funded startup falls into this category as well) or a much more conventional setup where the role of outside investors is taken by old-fashioned banks whose possible returns are strictly limited to what the credit contract says.