I would agree with your four choices, for sure. I would also add these:
They should have great body awareness. It seems obvious, but it’s never talked about. Unfortunately, I see so many trainers or fitness instructors who have poor form and little attention to detail or who just aren’t very athletic or coordinated.
Also, a great trainer will tailor the regimen to what each individual client needs, including different ways of communicating with each of them. Every single client is different. So while a tough approach might work well with a type A client, for example, more gentle encouragement may work better for others.
All of the above and:
Empathy – see the other point of view. You may not agree with it but you have to acknowledge its validity for the other person.
Be a good listener – you can not only read between the lines but also hear the true meaning under the words. Active listening is a skill that can be acquired through practice.
I agree with all previous posts. I would add:
1) A high level of honest and sincere communication skills (we deal with so many different people, in so many different environments, with little to no preparation. It really is good to know how to adapt your message and/or overall communication style on the fly, and still be able to develop fabulous rapport and connection).
2) Personal and professional integrity, respect, and ethics (kind of all the same thing, but still…). You have to present yourself in every situation in such a way that if somebody was watching you (and you didn’t know) you wouldn’t be embarrassed or immediately change the way you behaved.