It is an age-old distinction that often gets little practice and much lip service. There is supervision/management, and then there is leadership. No matter what job title, project managers can be either supervisors or leaders.
Supervising/managing simply means overseeing and directing work. If you have a deep understanding of PMBOK and other methodologies but still manage projects like a robot, you are not leading, but supervising. Another interpretation is that the tactical focus is primary and the long-term view secondary, if any.
Leadership is the ability to provide guidance and support employees in achieving their goals with less effort. It includes all elements such as training, example-setting and continuous improvement of systems. together. You don’t need to be charismatic to be a great leader. There are many other aspects to leadership than dynamism or likability. All of the above is required, but you must also have a long-term perspective that you use to view everything you do and all your teams do.
Organizations that want great project managers and leaders should train them to be great leaders. This is a correlation of Point 6. Leadership is an essential part of any training program. Leadership skills are essential for upper management. They must be lifelong learners and have a deep understanding of leadership philosophy. They should be passing that knowledge on to their project managers and other employees in an organized manner and on a regular schedule.