I am trying to set up an album where only members of a certain group are able to upload files. And this should be the only album where they are allowed to upload their own stuff. All of the other albums should not allow uploads.
Maybe I'm missing something really obvious, but I see no such feature in the album properties or group settings...
Groups control panel: set "Public albums upload" to "allowed" for that particular group. Don't worry, they can't do anything bad with this setting yet: it's a general toggle that allows them to upload to public albums, but by default all albums are set not to be uploaded to. So in a next step, you have to set up one album where the uploads can go to. Before doing that, go to config and make sure that you haven't set "Files and thumbnails advanced settings" -> "Albums can be private" to "No"; that setting must be set to "Yes".
Finally, go to the album properties screen of the album that is meant to be used for uploads for that group. Enable "Visitors can upload files" for that particular album and save your changes. That's it. Log out, log back in with the test user account that is member of said group and test things.