I think fidgeting with specifically a phone is the issue. It is unprofessional and disrespectful to make calls, answer calls, or text while seeing a client, and holding your phone makes it seem like she's on the verge of doing that and that therefore she is not fully present with her client. It's as though she's itching to communicate with someone else the whole time. Which is terrible because one goes to get an hour of undivided attention.

Fidgeting with a random item would be annoying I think, but probably tolerable.

I don't think your request that she put it down is at all unreasonable or excessively controlling.

What is MOST worrisome to me is the way she at first welcomed your feedback and commended you for making it and later was snippy, suggested that you were controlling and acted like the problem was yours to deal with. And she didn't even acknowledge the fact that she had changed her mind about the issue. That is crazy making. And, to my mind, absolutely not okay.