Final answer:
The correct command to promote a 'follower' node to 'master' or 'manager' in a Docker Swarm cluster is 'docker node update --role manager'. This grants the node managerial responsibilities within the swarm.
Step-by-step explanation:
The command that promotes a 'follower' node to a 'master' or 'manager' in a Docker Swarm cluster is b) docker node update --role manager. By executing this command, it updates the role of a specified node within the swarm to a manager role, allowing it to participate in the consensus and administrative tasks vital for swarm management. It is important to use this command with caution as having too many managers can negatively affect the swarm's performance and reliability.