The problem with getting this across is highlighted in one word from HWD's post above Panic. Can someone please explain the link between Mayday and Panic?
I hoped the single quotes would mean that I could get away with writing a whole paragraph explaining the difference between the process of assessing, diagnosing, resolving/mitigating and reviewing a potential emergency situation using the appropriate checklist and the instant canned drill response required in some few very specific scenarios.
Regardless of the opinions about what is or is not a Mayday in relation to an engine failure, the received training by people who make their living flying these things around Europe is not only specific but also totally logical. The name of the game is to give yourself room, keep as many options open as possible and account for the thing you didn't think of. As an engine is a critical component, there are potential ramifications that are not at first obvious. The absolutely last thing you need when single engine is to go-around or be generally delayed by traffic and other stuff. If you have told ATC that you are single engine, then I can't imagine that they would treat it as anything else, but it is probably be a mistake to assume that they will treat the situation as would expect unless you as commander unambiguously told them exactly what you want and expect.
So, if after trying to correct the situation, you are still with one engine, the initial Mayday will maximise the time available for everybody to ensure the above. As said by me and Foxy, downgrade if appropriate.