These days, people around are more likely to just video capture the emergency event instead of calling for help.... So may be the meaning of emergency-phone has shifted by now or soon will shift.
You're right there; "Smartphone" is becoming a bit of a misnomer now as the "phone" part is becoming very much secondary. Yes, it's not 1988 anymore and they are more mobile computers that can also do calls.
There was a recent case of a young lad who got stabbed in London. What did he do - call for an ambulance? No, he Facetimed his mum who had to work out where he was and she then called the emergency services who managed to get there in time to save him. Maybe I'm just getting old but 
After one faced a real medical emergency, one's perspective changes...
One danger is:
when one is hurt, one may not be in a good enough mental condition to understand the nature of the emergency. Particularly when the emergency involves blood lost or anything else that may impact mental acuity.
Normally, I rather not get so personal. I had my reservations about putting this in a public forum, but I rather fellow forum members not to have to face such "excitement" and also be aware how ones judgement can be impaired when one is in poor physical condition before having to face it.
One time, I made my wife go to the emergency room. She had out-patient tumor surgery earlier that day. I was not seeing what they told me to expect. By late evening, I begun taking photo's to measure progress/deterioration. By late night, I was alarmed but
she thought she was ok and just wanted to go to bed. Reviewing the pictures I took, I know she may be in more danger than she thinks, so I argue with her into submission. We got into the car and drove. I did not call emergency because I believed the time it took for the ambulance to pick her up and get back would have taken too much time.
All along the way to the emergency, I was thinking, there would be hell to pay if she was indeed AOK and I just made her go in the middle of the night.
She was well enough to walk in. After vital check in the emergency room, the medical tech reached for a big red alarm button on the wall. As the alarm started blasting, she got on her walkie-talkie and said something as she wheeled my wife into what they called the "resuscitation room" -- perhaps expecting she would need that any moment... It took a few hours to stabilize her and emergency surgery the following morning. She needed a couple more days in the hospital before she was well enough to be released.
My wife saw those photos I took that night a month or so later. She was shock that she refused to go to the emergency that night. That really strikes me how
blood lost can impair a person's judgement. That "night at the emergency" seer into my memory very deep.
It was that night I was thinking of when my phone was "upgrading..."