No, it ruled that they can be treated as persons. And everyone thought that that meant what you said. But a "person" is a human being with corporate responsibilities, and so had been given a different set of privileges from the corporation that they worked for. Citizens United ruled that or the purposes of political involvement, the the corporation can be given the same privileges as a person working in a corporate capacity.
Dense? Yep. But not tangled - very specific.
Thank you for the clarification.