I agree with several of the others that making sure the school fit is actually really working is the best place to start.
In general, kids do tend to let it out more at home then out in the world. It's actually a good thing since acting the same everywhere can be a sign of problems. It's good that your daughter can differentiate between different areas of her life and can control herself when she needs to (school), but relax that control when she is in a safe place (home). it's aslo good that she can use her words to tell you and UNDERSTANDS why she is acting this way. She is acting her age, but has the understanding of an older child. It's actually a good thing for you, believe it or not.
Since she already understands why this happens, odds are she probably doesn't enjoy it but doesn't know what else to do to feel better. When she isn't upset you could try sitting down with her and discussing that aspect of it. Ask her what she thinks might help her. Let her use her mind to help balance out her physical age. Enlist her help in solving the problem. If she has no suggestions, offer some like a non-punishment time out (in other words, "I'm going to me room LEAVE ME ALONE!") and then coming out when she has calmed down some, punching a pillow, jumping on a trampoline, running around the yard, screaming into a pillow, throwing ice into the tub/shower (trust me, this one is GREAT!), etc... Offer to help her notice when things are spiraling down, like I think you might consider taking a break before things get too hard to handle.
Have a family safe word that anyone can use and that everyone must respect. Use it only when things are getting bad enough that unacceptable responses will occur. It has to a be a word that you will remember and not use under normal circumstances. When someone uses it everyone has to step back from whatever the situation is and go take a ten minute (or more) break from one another. When it is over you talk about why it got so bad, NOT what the problem was to begin with. Once that is sorted out you go respectfully back to the first issue.
Use a talking stick. Only the person with the stick can talk and the other people involved have to listen respectfully. It's amazing what a difference this makes.
All of that said, talking with a professional isn't a bad idea. However, be wary of diagnoses that don't ring true. There are some people out there who will diagnose whatever the issue in style in anyone who they see. Be wary of doctors that insist on meds off the bat without other options being tried first or concurrently (with depression that would be talk therapy and a consistent exercise schedule). In fact any doctor who wants your kid on meds without talk therapy as an option is not one I would trust... Also if drugs ever are prescribed make sure to ask about the history of their use with children your age and slightly older. Several drugs act very differently with kids and can actually exacerbate problems.