But can't a doc insist on a hospital stay following the emergency, which then isn't covered by insurance as an emergency?

By "insist," I mean that leaving would be against medical advice -- and I know some insurance will refuse to pay for the preceding stay.

I wish this stuff weren't so hard to make sense of.

I don’t think so, if you get discharged from the hospital under your own recognizance, you are no longer in an emergency. If you are not discharged, then how could they argue it is not an emergency?

Who knows how well it works in practice.