I don't think that means what you think it means, perhaps you are international? A Public company is one that is owned by shareholders of a public exchange. This is why Elon can buy into it and become a member of the board. If it were a private company he could not do that.
No, it is literally a public company. TWTR is traded on NASDAQ. Which is why we are having this discussion based off of SEC filings. Private companies are not required to file with the SEC.
Edit: sorry, I mixed up public company and public good. Twitter is a public company, but not a public good.