There's no real excuse for non-standard RT terminology in a case such as this one. With 'Mayday, Mayday, Mayday,' then you have made it clear that you have declared an emergency; everything else comes from that understanding, particularly that other traffic knows to keep silent so that ATC can communicate clearly with the aircraft in distress. If I hear a 'Mayday' I know to pipe down to let the crew and ATC sort out what should happen next, or perhaps pass a message if the call seems to have been missed. Here we can see that no other crew pointed out the missed emergency call to ATC; that seems to show that nobody else understood it as a 'Mayday' call, showing this is not some argument about how 'we' do it compared to 'them.'
I have noticed that the USA seems to have a lot of non-standard RT. What some here have noted with a certain amount of pride, that we have most of the traffic, really is part of the problem, that we sometimes forget the standard procedures, as if we all should know what we mean anyway. 'Roll the trucks' sounds like something from a movie, Die Hard VII, perhaps. Coming from an aircraft in flight it's almost meaningless.
I got my American radio licence by sending in a postcard. That was it!
I got my UK licence by taking two short writtens, one for IFR and one for VFR. (Brits speak completely differently when the flight rules change. This is something that most people do not know.) Then I did a two-way oral test, including making a proper 'Mayday' call.
I got my German AZF by taking a spoken test in German, since they graciously accepted that I already could communicate in English, when that included a whole lot of possible situations. It was that typical German thoroughness, with the testing being done by the Federal Ministry for Post and Telecommunications. (I now have a small blood-group tattoo high up inside my left arm to show my status post-test, plus of course I had to swear the oath.)
Some of that might account for differences one can easily hear between different RT standards in the States versus elsewhere. We usually get the job done well enough here, but sometimes we need a higher standard.