May I suggest “community software”?
“Public software” could potentially be fudged to apply to anything that anyone has access to. It’s not your intent, but it would be easy to misinterpret Facebook as public software because it’s not limited to a particular group from an end-user perspective.
Community software suggests that it’s built as part of a commons. No, there’s no license particularly associated with that — although that could have interesting unintended side effects down the line — but you still have the expectation of not just sharing publicly, but potentially taking feedback and pull requests from other users.
I suspect there will be a lot of pushback from the free software folks in particular, who were already uncomfortable with “open source”. But honestly, that term (and the overloaded definition of “free”) has become so problematic, and so closely tied to a specific political movement, to be usable in many situations. In my view, it tries to change too much, and espouses an idealistic worldview rather than an incremental improvement.
In turn, I think community software is better than open source because it talks about the process of development, rather than simply a right that you have.