Personal trainer/coach, to me are the same thing.
Long time ago on this forum someone tried to insult me, he said "a personal trainer gets the first 50% of someone's possible performance, a coach gets the last 10%", or something similar to that. He later apologised - but he was right.
Take some 30yo 80kg accountant. World record squat and 5km run time for him are something like 320kg and 13'. If you get him to squat 160kg and run 5km in 26', you've totally changed his life. 50% of possible performance is actually pretty good for someone who just wants to live a good life. To get much past there he's going to have to focus; not many people will be able to do 75% in two or more things at once, that'd be squatting 240 and running 5km in 19'. And even just looking at one physical quality, not many people with full-time jobs, families and so on are going to go a lot past 50% WR in anything.
A coach is for someone who's a competitive athlete. And a competitive athlete is going to be past 50% of gender/age/weight-matched WR. But you don't need to be at 75+% of WR for health - you do need to be above 25% though, that's for sure. I find 50% is a good mark. I've told my intern I won't give her a reference until she can total 50% of WR.
So a personal trainer prepares you for life, a coach prepares you for a sport.
And whatever you guys say hypothetically, I can tell you that nobody
who actually pays for training cares what trainers and coaches look like. I mean, being morbidly obese would hold you back, but that applies for any job, let's be honest.
Nonetheless, I think the trainer or coach needs to at some point
have been through the process of training. They need to have gone far enough in something that they got stuck, and had to work out how to get unstuck. They need to have trained hard enough that it was a challenge to balance training with the rest of their lives. This process will teach them things about training which they can apply to clients and athletes, and perhaps more importantly give them that balance of empathy and hard-headedness a competent trainer or coach needs.