You appear in public. You buy things in public. You might sell things in public, or work in public. Although your actions can be described as public in some ways, you are still a private individual.

Same thing with companies: private companies that are publicly traded. Amtrak is a public company because it is owned by the government. Coca-cola is a private company, even though its stock can be purchased by any member of the public.

Whether Twitter operates as a public square or not is a separate question, which the courts are looking at. Although this court isn't looking at it any more.