As an intern undertaking my first industry-relevant experience, I often find myself drawn to continue working on projects beyond my scheduled work hours. Should this extra time be considered overtime and discussed with my manager, or is it expected to be unpaid and regarded as a personal choice?
That is very dependent on your company and team. You should speak with your manager. Since interns are already a net negative financially for most teams, it's rare to have the budget to provide overtime or for it to make sense to do so.
Where it gets tricky is that, legally and ethically, you should bill for any time you are working when you are hourly. However, if the decision is not being dictated by the project needs and your manager, then you are choosing to work longer for your learning. That's more your choice and could be considered outside of work. I wouldn't work longer hours and bill for them without talking to your manager. Doubly so if you are doing it to maximize your learning versus responding to project requirements.
It's very rare to have interns be eligible for overtime pay - I have worked with hundreds of interns, and I don't think I have ever seen it apply to them.
And for interns, this makes sense. The vast majority of interns work on self-contained projects where nothing is time-sensitive - Interns are pretty much never placed on something like a 24/7 oncall where you are required to work >40 hours per week. You just need to get your intern project(s) done before your internship ends, so the challenge of your internship reduces down to how you can get maximum value from the ~40 hours a week you work.
In your scenario, it seems like you're working extra hours to expedite your project progress, learn more, and exceed expectations. This is 100% okay (most interns at Meta did this), but that's your choice so the employer has no responsibility to compensate you extra for it.
When it comes to getting maximum value from your time, I recommend these: