Deelilah, your thinking is wise and I would only share what you are comfortable sharing, it's not up to you to make sure the counselors get the correct information.
I worked at a rehab for 2 years and the counselors can spot BS a mile away, they've heard all the stories and blame and shame and in the end they are well trained to deal with this and make progress with the resident who usually loosens up once they have been there a while. The rehab I worked at was a 9 month to a year program, had excellent mental health professionals available to deal with some sensitive issues (like childhood sexual abuse) and counselors available to talk to 24 hours a day, as well as 12-step meetings each day and house life skills responsibilities like cooking and laundry and cleaning.
I hope rehab helps him, it's very early for him and there will be lots of changes over time.
Take good care of you, do what is comfortable for you, and I think you will get through this in a healthy way.
Hugs