I think it depends on the counselor.
The main thing I can tell you is that if you feel that the counselor is not the right fit, make sure to find another. The counselor we saw after the first dday seemed very unexperienced and very "cookie cutter". We only saw her a couple times because we were getting nowhere. I then went to IC and found that the counselor seemed to have her mind elsewhere during our sessions. So..two years later I learn that he was still hot and heavy with the OW.
I also agree with the other poster that having a "list" of concerns is a good idea because things can get very emotionally tense in MC.
I really liked the promise your MC had your H make, so I had my DH get down on one knee last night (he never actually did that when he proposed!) and make me the same promise. I could see the severity of it hit him, but he did look in my eyes and make that promise.
I followed up by asking him if there was anything he wanted me to promise. He said there wasn't, but I think it seemed to mean a lot to him that I would ask. The rebuilding of this marriage, after all, will take both of us to achieve.