There is a linguistic problem here. Private sector is understood differently in the US than almost everywhere else.
There may indeed by a linguistic problem as you suggest, since if just one person understands a term-of-art different to the rest, as you state you do, confusion inevitably obtains. But there is not the widespread confusion you imply. I am oustide the US. I do not understand the term differently. "Private sector" is one of those terms which (like when lawyers use the terms "convention" or "common law") has multiple subtly different meanings which are wholly dependent upon context. In British English, "private sector" means, pretty much, the same as it mean in American English. Its direct antonym is "public sector". It therefore means exactly what it says on the tin. In the debate over socialism and communism in late 19th and most of the 20th Century, the controversy over ownership of capital and the means of production was between the public sector ("nationalisation") and the private sector ("privatised business") It is true that often "private sector" is used to refer to a for-profit structure for business ownership being, e.g., better than a bureaucratic socialist model (vide the - mostly - successful privatisations by British Governments of both political colours in the 1980s and 1990s). The expression used in the UK when you want to distinguish this kind of private sector body from the others is "voluntary sector" or "third sector". This means charities, non-profits and NGOs. But you really are dancing on the head of a pin here. ICANN is not private sector since it has governments as a fundamental and important part. It also is not private sector since it has the voluntary sector as a fundamental and important part. But it is undeniably "rooted" in the private sector irrespective of which definition you use. It has been quite fashionable since the aforementioned privatisations in the UK for the government to get outof the business of regulation. Hence the rise of "self-regulation" in the private sector. An example of this is the Phone Paid Service Authority (originally ICSTIS) http://psauthority.org.uk/ - a self-regulatory authority that is wholly owned by the the private sector that it regulates. (Yes, the poachers DO appoint the gamekeeper). ICANN is different from this exactly because of the participation of the public sector (GAC) and the voluntary sector (ALAC). But ICANN is still a private sector organisation. And even it, in some contexts you want to distinguish the type of organisation that ICANN is from a pure private sector organisation, we have coined the term 'multistakeholder' for that purpose. That is our collective contribution to the language! Its direct antonym is "multilateral" which, it seems to me, you want, and I don't.