Earlier quoted context omitted.
No, your understanding of it is wrong. Trump is limited from blocking people because he's the President. If he were to resign tomorrow and go back to being a private citizen, he could block away to his heart-s content. Twitter, the company, can do what it likes because it is not an organ of the government, notwithstanding the choice of some government officials to make use of it.
But you are missing the part where Trump is allowed to pick the private company, and is thus able to pick one which enforces a ToS that the government would not be able to enforce themselves. This creates a loophole where by the government only has to pick private companies that ban anyone who is against the current government to get a defacto censor on the public square that wouldn't otherwise be allowed.
You're overlooking the fact that requiring Twitter to retroactively alter its own terms of service once it provided services to the government would be a violation of the takings clause. In your hypothetical example, it's more likely that a court would enjoin the government official from use of the company with the overly restrictive ToS for official business, rather than forcing the firm to alter its policy.