Even as an intern, you will have full ownership of your project. You may have other projects to support, but every intern has a capstone project they are responsible for. This might seem weird, but you may find yourself in a meeting where you have the most knowledge on a design, and your input can guide the direction of the project. Things might get confusing with so much freedom, but your manager serves as a mentor and provides guidance so you'll always be in a position to succeed.
Ultimately your test lead approves your final reporting and your manager is responsible for the quality of the work your team puts out. You are still responsible for all test and projects you manage, and are expected to be able to update all teams on the status of a project weekly. This helps everyone stay on the same page, react if there is risk to there own projects, and offer guidance if needed.
At Apple, everyone take complete ownership of the work that they do. This means that their development lies in their own hands and independent ambition is required for growth. There are many opportunities, but none will be instantly provided unless one has demonstrated the desire and competency. One of the many wonderful things about Apple is that your team is always there to help you when you ask for it, so there are always resources available.