Well... yeah? That's describing the event horizon. It's a term roughly as widely used as "singularity".

Talking about the inside of a black hole is indeed rather pop-misunderstood though, yes. But it's not like physicists are especially confident about the details either. Theoretical astrophysics changes a lot as time goes on and our instruments improve, and it's a rather hard field to do experiments on to get better data quicker.

No, sorry, the singular surface in a Schwarzschild spacetime is not the event horizon. Nothing particularly interesting (from the GR point of view) happens at the event horizon.