There is a very simple answer to all of this. Don't fly for a company who only have aeroplanes with two engines!
I managed that for 46 years and never flew a commercial twin.
Therefore an engine failure was but a mere annoyance and a Pan would suffice on most occasions.
However, had I been put in the unfortunate position of having to make do with just two engines, I think a Mayday would at least have been on my horizon if one of them failed.
(Come to think of it, just getting airborne on two engines might have been getting close to an emergency!)
In my last company the SOPs called for a Pan with the loss of one engine and a Mayday with the loss of two engines.