I had an informal interview for a postdoc position that was followed by a formal one. Both went really well (at least in my opinion) and the formal interview exceeded its specified time because of the interest they showed in the work I presented. Upon asking, I was told that the process may take up to 2-3 weeks, speaking of which, there is no official/unofficial word about the outcome of the interview and more than two weeks have already passed. Should I wait a bit longer so that I don't come across as too desperate?
I think this might depend on whether you have other options that you need to pursue soon, or just want to quiet your fears. In the first case, you should probably ask so that your options become clearer. But in the second, I'd wait, perhaps even after the 3 week window ends.
I think that if things were bad you'd hear sooner, so it seems that you are still in contention, but there is no way to judge your relative position among finalists.
I would recommend, at the moment, that you wait, but don't neglect other options. But the only information I have is what you wrote. You may have more.
Not to mess with your mind, but it is possible that they have made an offer to another person and are waiting for acceptance/rejection before contacting you again. That would push the time frame out a bit. If that is the case then you aren't going to get much useful feedback from an inquiry now.