I'm not in any major premed clubs, or even other ones either, so my leadership experiences don't come from there.
However, there is always something to do in your community for you to prove your leadership skills, even if it is fundraising for a cause/organization that sparks your interest. Once you do that, you are taking the initiative so that organization will probably ask you to take on more responsibility---> Leadership position. First find out what you're interested in. Then pursue organizations that work in that field, asking them what needs to be done to further the cause. They probably will not turn you away and if you prove yourself to them, and ask for a leadership position, you have now found something that you are interested in and also something that you have taken extra initiative to attain.
Many of my activities were not related to medicine either. This can actually be beneficial to you. In fact, the ones that interviewers have commented on most have been the non-medically related ones. Of course, make sure you do get clinical exposure elsewhere for balance, but don't worry about what it will look like on you application.
Do things because YOU feel interested in them and YOU want to do them. If you like what you're doing, you will naturally excel at it anyway.
