I flew a SAR helo at a fighter base for several years and whenever the fighter guys had a 'problem' they were very reluctant to declare an emergency using the standard phrases of MAYDAY or PAN but generally just said they were declaring an elergency.
They few times over the years I've had a problem myself and needed assistance I've declared the appropriate phase and the required help was forthcoming very quickly.
I think there is a reluctance to use the standard emergency phrases as they are used very infrequently and are unfamiliar to most pilots, a similar thing happens when a standard phraseology is changed for whatever reason until the players on both sides of the microphone get used to it and it becomes commonplace.
Regards,
BH.