In my modest opinion, a real personal trainer is someone with a degree in exercise science and a certification from ACE or ACSM. However, someone with the ACE or ACSM certification, who has a degree in something else, is also qualified in my book. That doesn't mean, of course, that someone with these qualifications will not end up peddling some weird "diet" or training plan that doesn't make sense.
Although I recently let my gym membership lapse, I have spent about 25 years hanging out at various gyms, as both an employee and a user. The field has evolved slightly since the mid eighties when I got my group exercise certification from ACE; back then, just about anyone could walk in and call themselves a trainer. At my last club, all of the trainers had degrees in exercise science. Many of them had private clients outside of the ones they worked with at the gym. They knew what they were doing. I worked with one for about a year, before my cycling trip to Austria and I never felt like he was doing anything contraindicated or giving me a hard sell. Of course, there was a price tag for this, that a lot of people would not or could not pay.
Don't get me started on the certified spin instructors who continue to do the push ups on the bike and the crazy rpms. While I don't love riding my bike on the trainer, I don't miss the characters and weird stuff that went on in spin class.