The dirty little secret in CS is that unit, functional, and end-to-end tests are all the exact same thing. Watch next time someone tries to come up with definitions to separate them and you'll soon notice that they didn't actually find a difference or they invent some kind of imagined way of testing that serves no purpose and nobody would ever do.
Regardless, even if you want to believe there is a difference, the advice above isn't invalidated by any of them. It is only saying test the visible, public interface. In fact, the good testing frameworks out there even enforce that — producing compiler errors if you try to violate it.
