How do I track running, as opposed to walking, in the iPhone Fitness App?

Yes. By default, all walking/running is recorded as "Walking + Running Distance" in the Health app and this is also the distance you see in the Fitness app.

And yes, the data is already in your phone, so it would be nice if the software was smart enough to infer when you're running vs. when you're walking. Thats just something they haven't added (yet).

I'd like to clarify though, that - even if you have an Apple Watch - you would still need to manually click a button to start a workout. The watch automatically tracks minutes of exercise and calories burned, but it does not automatically classify anything as a running (or walking) workout. For that, you always need some activity tracker app.

On the Apple Watch, that app is built in, and it will even automatically detect you are running or walking and ask if you want to track it as a workout. But still, you always need to confirm which kind of exercise you are doing before or while you are doing it.

On iPhone, there is no built-in activity tracker, so you would need a third party activity tracker to have that data in the Fitness and Health apps.

Hi!

To track your runs in the fitness app, you can use and Apple Watch or a third-party activity tracker app.

This page tells you how: Sync a third-party workout app to Fitness on iPhone - Apple Support