Alright. After hours of trying, failing, searching, re-configuring and finally wiping my TLA website setup, so I could test a clean one identical to this. I finally found where the problem is. It's in the JFusion <--> DokuWiki 'Authentication Mod', which is used to enable reversed dual login. Meaning it's the bit that enables the auto log-in between the main Joomla site and the Wiki.
The fix is to disable this module, and just live with, (for now) that log-in states between the Joomla and DokuWiki is not carried over. You need to manually log in and out of the wiki.
I will keep this issue on my
****-I'd-like-to-fix list, but it will not get high priority right now.
Using MediaWiki was a no go. I get an issue with JFusion if I enable it on MediaWiki. The issue prevents me from getting to the page where I can configure the JFusion plug-ins. It just gives an error message and I need to remove the wiki install to get rid of it.
I'll keep this on my previous mentioned list too, but again, no high priority for this now.
So, to summarize: The wiki is working. You can read it even if you're not logged in. You only need to register at the front to get a wiki account, but you need to log in/out from the wiki it self.