If 'open source' isn't about having.... I dunno, all your source-code being licensed under an OSS license... then what exactly is it?

I think the word you're having more of an issue with isn't 'open source', but defining what exactly a 'project' is vs. a business. On that latter point, I would agree with you that a project doesn't seem as much of a 'project' when most of the core people are employed to work on it towards specific goals the funders have designated.