No you are NOT correct.
He belongs to a professional organization that sets standards of behavior for its members. He represents himself as a member of that organization when he makes these proclamations. The organization has told him to stop. He did not, so the organization told him to take training. Peterson claimed that he had the right to say these things. The court said he had the right to say these things as a private citizen, but the professional organization has the right to set standards for its members. Peterson defied these standards, so the court ruled in the organization’s favor.
If you break a contract with someone and the court rules against you, that’s not the government imposing its will on you. It’s you breaking the terms of the contract and the court enforcing the terms of the contract.
But such nuance is irrelevant in The Forever Culture War Outrage Machine.