I don't necessarily see a direct relation with "open source" and "private" (even though apps which focus on one usually also focus on the other)...
So I think you're asking two very different questions:
- Are we being loose with the term "open source" [...]?
- Are we being loose with the term "private" [...]?
Server run by the main developers of the project It is not focused on any particular niche interest - everyone is welcome as long as you follow our code of conduct!