How do I state that the reason for leaving the last job was a mutual miscommunication with the management?

We parted our ways because both sides had different expectations from the role that I took. We parted on good terms and the (former) project manager even said that I could give out her phone and she can commend me as a worker.

How do I declare that fact during the interview when asked? I've worked there for just a few weeks, so it's always brought up. It feels like if I'm not careful with the words that I choose it may look like I got fired for being incompetent or just bad.

How do I declare that fact during the interview when asked?

Must note that you should mention that not when, but if asked.

No need to elaborate much on it, in fact your current phrasing on this post reads good to me: Mention that you parted ways because both sides had different expectations from the role, but that you ended up in good terms.

You could also consider including your former manager's phone somewhere on the part where you describe that past job experience on your CV, given that she said it's ok for you to share in case they need a reference.

Update: Reading more closely, seems that you worked there for just a few weeks. In that case, given it was such a short experience, I concur with Justin Cave and would not advice including it on your resume.

Not only it was a really short time, in which you perhaps didn't learn/do much, but including it may raise more eyebrows and questions than have a positive impact on your chances of landing a new job.