I think you are misinterpreting the advice on brevity. You can be thorough and cover the relevant issues concisely. It is not a contradiction. Go ahead and write it with all the details you want.
Your appeal really isn’t the issue. This forum sometimes offers the opinion that some time off from school might be a good idea.
You already got some stability with meds but still had academic issues, mainly, perhaps, from commuting.
In your letter, certainly include what your wrote in your first post here, with the timeline and problems experienced.
But the most important thing is to present and focus on the solutions that might argue that you are able to be successful.
- You have a diagnosis and medications that are working without side effects.
- You are working with a psychiatrist and therapist
- You have housing near campus (or are classes online?)
- You have been honest with your parents and have their support
- you have clear academic goals and will use any help available including advising and tutoring as needed.
You can’t just promise these things but need to have concrete proof.
You can also ask for a leave to be considered and promise to meet criteria for returning. Perhaps they will agree to that rather than dismissal.