You may well be right. It contradicts other parts of the manual as it states:
One pilot amongst the flight crew, qualified as a pilot-in-command in accordance with the requirements governing Flight Crew Licences, is designated as the commander who may delegate the conduct of the flight to another suitably qualified pilot;
Commander - The pilot designated by the operator being in command and charged with the safe conduct of a flight.
You can have 2 PICs but only one Commander, so whoever signs for the aircraft has to remain commander unless we are planning in-flight maintenance checks. By the whim of Thor, what have I said!