The correct answer is B) Designated person in charge. The designated person in charge is responsible for appointing persons to be in command of the lifeboats and/or liferafts on a mobile offshore drilling unit. This is in accordance with regulations governing the safety equipment and procedures for mobile offshore drilling units. The designated person in charge is typically a senior crew member or safety officer who is responsible for overseeing the emergency response plan and ensuring the proper manning and operation of all lifesaving appliances. The other options are incorrect because the tool pusher, company man, and rig superintendent are not the designated authorities for appointing lifeboat/liferaft commanders on a mobile offshore drilling unit. Their roles and responsibilities are focused on other aspects of the drilling operation and not the specific emergency response procedures.