I think there is a slight difference between an engine failure and a precautionary shutdown.
No there is not when taliking about a twin.
Flying on one engine is flying on one engine, it doesn't matter how you got there.
Also a slight difference between and an emergency and an abnormal.
Absolutely right there is. As far as I am aware any loss of a system where the subsequent loss of a similar system would cause mortal danger is an emergency. This is not open to interpretation.
This is why on a twin 330/777/767 the loss of one engine requires you to land at the nearest suitable airport. Yes you can continue somewhere else but the guidelines become pretty strict.