Earlier quoted context omitted.
EMERGENCY- A distress or an urgency condition. DISTRESS (MAYDAY)- A condition of being threatened by serious and/or imminent danger and of requiring immediate assistance. URGENCY (PAN-PAN)- A condition of being concerned about safety and of requiring timely but not immediate assistance; a potential distress condition.
“Declaring an emergency” is US specific. The rest of the world doesn’t recognise this phrase you have to use either “Mayday” for immediate danger to life or “Pan Pan” for less severe problems. As I understand it, “Declaring an emergency” is the equivalent of a Mayday and the US is very slowly switching away from this non standard phraseology.
That said, controllers are human. As long as you get the message across somehow they'll do everything they can to help, even with non-standard phraseology.
Of course, the risk of non-standard phraseology is that you might be misunderstood - especially in countries where English isn't the primary language. It's still good to stick to standard ICAO phraseology whenever possible.