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.