How do you know if you're a leader? If someone is following. Laugh, but that's the truest measurement of leadership.
To truly lead, you need to be going someplace and you need to take people with you. If you're going someplace alone, that's exploration. If you're in charge of people but leaving them where they are, that's supervision.
Supervision is the day-to-day oversight of people. It's making sure they come to work, making sure they do what they're supposed to, making sure they have what they need. Supervision involves time cards and vacations and breaks and the order work gets done.
Leadership results in movement, change. A leader sees a better place in terms of what can be and how it can be done. Leaders see the vision, communicate it, and inspire and enable others to get there.
Often you will have to both lead and supervise, but keep the functions clear in your mind. You can lead far more people than you can supervise. I supervise 14, which is way too many. I lead a couple hundred.
If you do both, make sure you don't get bogged down in supervision. That's important too, but leadership is where the magic happens.