The qualifications for training people in exercise activities are quite varied. You could add to the list above: gym instructor, athletics coach, aerobics coach, physical therapist and many more niche qualifications that would suggest good value for what you want to achieve with your fitness program.
Sometimes it just boils down to how good the guy or gal is in an all-round sense. Having knowledge is not necessarily the same as being able to coach or teach well. We've probably all had experience of that at school and beyond. Even so, having qualifications should at least ensure that the person has a fundamental knowledge of the skills he or she is teaching. And while this is a useful start, there are many street-smart trainers without qualifications who do a great job.
Personal rapport with the trainer is also very important. It won't work unless the both of you can cooperate at a useful level.
Being an exercise physiologist or physical therapist may suggest a superior level of knowledge, but is it in the area that will be of most benefit to you? If you want someone to teach you bodybuilding or powerlifting for example, unless the exercise physio or physical therapist has experience with weights other than in a rehabilitation program, they may not be the best option. You are more likely to get better value from someone who has less theoretical knowledge and more practical knowledge of what works best under the iron. And when it comes to diet and nutrition, often all bets are off. Unless you employ a trained dietitian or nutritionist, you could get anything in the way of advice -- from useful to downright dangerous.
So the best advice I can give if you are looking to employ a trainer is to chase around and get opinions from people, talk to prospects, get references and read up first and know a few things to ask before you take the plunge.