As someone who over the course of 25 years has had more than their fair share of situations that required the emergency checklist, both operating/registered in the EU and the US, this is what I did and how it worked out:
I declared an emergency (Mayday x 3, or in the US "declaring an emergency").
We did what we needed to do.
We told ATC what we did/needed.
We landed safely.
We filed a report.
Never had any questions about doing it that way. I know it is only a sample of just over 10, but still...