From my reading, you are going from App-first to Browser-first, and less so from mobile-first to web-first.
I think it is critical make the subtle distinctions. We are in the business of serving doctors. For most doctors, having a native app makes very little sense. And yet, having a responsive mobile-friendly website is becoming more important than the desktop websites they presently have.
I agree that it is critical, but don't think it is actually a subtle distinction. It is a huge, glaringly obvious distinction. Which makes it all the more unfortunate that so many people reinforce incorrect terminology. I find it hard to believe people seriously don't realize their phones can access web pages, and that web pages can be designed correctly so they are accessible on all platforms.
I think it is critical make the subtle distinctions. We are in the business of serving doctors. For most doctors, having a native app makes very little sense. And yet, having a responsive mobile-friendly website is becoming more important than the desktop websites they presently have.