Unfortunately, leadership as it applies to medical school applications and 'real' leadership often tend to be two very different things. The leadership experiences listed on med school apps usually simply reflect some ability in, or at least familiarity with, managing others (as explained by
@karayaa). Fortunately, the experiences and lessons of real leadership are more personally rewarding and beneficial to you.
Truth is, real leadership does not have to be exceptional or lead to major life changing outcomes. This expectation tends to cause people to be afraid to step up to the helm of leadership. Good leadership is essential in almost any group situation whether it's a trivial task or a life-changing goal. And in my opinion, to be a good leader one simply must have the courage to
take responsibility for guiding a group towards a common goal. To be a great leader, one must seek to
put everyone else's needs in the group above their own while doing so. Most of what your mentor was talking about regarding leadership reflects the need for a person to put away their own selfish desires and fears so that they focus on serving those that they lead.
Contribution alone is not enough for leadership. A leader's job is to maximize the contributions of others towards the goals of the group.
While the pursuit of becoming a great leader may not directly affect your med school application, the benefits to your life may be immense. It seems that real leaders are sometimes hard to find (especially those that aren't leading solely for their own benefit) and the world definitely needs great leaders. I would have to disagree with
@karayaa about the lack of leadership roles in healthcare. In my experience I have seen many facets of healthcare which need to operate as a team. Multi-system trauma care in the ED or a code definitely needs a calm and cool-headed leader to guide the other members of the healthcare team in caring for the patient. While the doctor will naturally take the lead in these situations, there is a difference between managing a team and leading them and good leadership foundations could be key in saving a life.
Leadership usually takes time to learn and develop. The simple fact you are interested in leadership is a good sign you're on the right path. Good luck in this pursuit.