Public is being used to clarify your misunderstanding of what you copied in.
You said the Packers aren't really non profit because they have owners.
The article said they do not have private owners.
Again this is what the entire debacle has been about, and if you reread you will see this.
Wikipedia's opinion, or the correctness of it is not at issue. Your statement that they aren't really for profit or non profit because they have owners (evidently your reason they are not non profit) but they do not benefit financially (evidently your reason they are not for profit) was a misunderstanding of what you read and copied in.
That is all it was. Reread please.
Since you keep quoting the expression "private owners" perhaps you could explain what you think it means. I have suggested rewording the sentence a couple of times in line with what I think it means, and you keep going back to "it says private owners."
Ownership is the quantitative difference between for- and not-for-profit organizations. For-profit organizations can be privately owned and may re-distribute taxable wealth to employees and shareholders. By contrast, not-for-profit organizations do not have private owners. They have controlling members or boards, but these people cannot sell their shares to others or personally benefit in any taxable way.
The terms "privately owned" and "private owners" are in parallel and they should mean the same thing.
In this context, the authors of the article are not making a distinction between a privately held company and a publicly traded company. They are making a distinction between owners (whether one or thousands of them) who own something as individuals (in their own right) and the "owners" of a nonprofit organization who as a group control the organization but do not "own" it as individuals.
Perhaps a specific example would help. Harvard College, which curiously includes the rest of the university, is "owned" by the Harvard Corporation which is the President and Fellows of Harvard College. Until recently, this was a group of seven people. It is now thirteen people. They do not "own" Harvard College as individuals, i.e. as private owners; they "own" it as a group. As the article said, "By contrast, not-for-profit organizations do not have private owners. They have controlling members or boards..."
Technically, Harvard College has owners, but the term "owners" means something so different in that context from its common usage that it is not usually used.
A person who owns one share of Microsoft stock owns it as an individual, i.e. as a private owner.
I said that the Packers do not seem to fit the category of a nonprofit organization (and that is all that I said) because they do in fact have "private owners." The Packers sold stock, and people became owners by buying that stock. They own that stock as individuals (in their own right). The people who bought that stock own the Packers in the same way that the people who buy Microsoft stock own Microsoft.
Microsoft has "private owners," and the Packers have "private owners." Harvard College does not have "private owners."
There is, by the way, no sense in which Harvard College, for example, is "publicly" owned, and your using the term just adds confusion because the usual distinction is between a public company and a private company, i.e. whether or not the company is publicly traded.