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.)
Some webhosting companies already provide the entire combination of PHP + database + forum, for the price of webhosting. (At least this was the situation a decade ago.) Then you just set the access rights to "only registered and approved users can read and write" and you're ready.
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.
Zulip, Discord and Slack are all options as well
However, these are all very bad for searchability, archiving, multimedia content, and creation of permanent content of any sort.
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.