My two cents:
When I ask people to post in a specific category I tend to provide a link to the category to avoid any confusion. (That’s regardless of whether the category is public or not). With some people, I even include a link that opens the editor and starts a new message for them)
Tobias has it in his screenhots but doesn’t mention it in the text: “spaces” gives a similar sense of space (duh!) as forum, while avoiding the abuse Hawk mentions.
I don’t think you’re overthinking this (it depends on your users, and you know your users best). Now, my first thought is, of course, that PMs are officially no longer called private messages but personal messages. So one way of avoiding confusion in the long term would be to promote the term personal messages. But these kinds of cultural changes are difficult to achieve.
So if we stick to the term private, then I tend to use the terms private, semi-public and public, to indicate the three levels of visibility or access. I agree that “semi-public category” sounds a bit bulk, but the question is: do you need to use that term at all? For example, in the category description, you can say “This category is semi-public.”