MyBB (or similar) with a custom theme.
Aesthetic: lots of themes available, and making your own seems easy.
Inexpensive: can’t beat “free” for the software, and cheap hosting that supports PHP+MySQL is plentiful.
Private: trivial to set up basically arbitrary access controls, as with any half-decent forum software.
Easily set up: standard PHP+MySQL stuff.
(I strongly anti-recommend Discourse as a forum platform.)
Everyone who participates probably isn't a github-using programmer, but if they were, a stupid five-minute solution might be to just set up a private github project and use its issue tracker for forum threads.
I don't know what "private" means to you, but if you just mean you can control who joins, I think google groups are a good choice for 2 - 4.
Zulip, Discord and Slack are all options as well, though they all (to differing degrees) encourage shorter, chattier posts.
I have used a number of discourse forums and they just feel bad/wrong but I cannot explain why. I would also vote for more of an old-fashioned php BB with a nice theme. Those are always great, even though all my intuitions tell me they seem like they should suck. Shows how little I know.
Eg https://github.com/phpbb/phpbb
Also has styles: https://www.phpbb.com/customise/db/styles/board_styles-12?sid=6245508b90fd3410be19888406fae215
Basically I'm repeating what Said said
Reason for Writing: I want to gather information for a friend who is a professional in the behavioral sciences.
Epistemic Status: Less than 2 hours googling how to build a forum, and a couple youtube videos
Requirements:
The goal is to create a small, private forum for discussions.
I've looked into discourse, not sure if that's the best option.