You cannot have a % ownership in a nonprofit because its resources must be used exclusively to carry out its mission. You could have a % control in its decision making process.
And businesses with owners can be dedicated to their mission.
And public businesses are run by a board of directors.
The main difference isn't what you're saying, which is somewhat subject to interpretation. It's just that people can't invest money with expectation of a return, other than via employment, speaking fees, etc etc.
For an example of the former, the previous head head of Mozilla received a compensation that rose from about 2 million a year to nearly 7 million a year following hundreds of layoffs due to declining revenues. For an example of the latter, following the earthquake in Haiti, the American Red Cross raised nearly half a billion dollars. After all was said and done, they built a total of 6 homes. [1]
Basically, non-profit is a tax-status with conditions. But those conditions are sufficiently unenforceable or side steppable that it's ultimately just a tax status. And the whole game of OpenAI being nonprofit until profits started rolling in is just making this even more clear.
[1] - https://www.propublica.org/article/how-the-red-cross-raised-...