Thanks for reporting the issue to LJ, guys! markf (one of the LJ staff members) has messaged me with the following solution:
I've seen a few reports about OpenID problems based on your post at http://pkmncollectors.livejournal.com/1
4431841.html, so I spent some time digging into this today and found the cause. The problem occurs if an account is marked as having adult concepts on http://www.livejournal.com/manage/setti ngs/?cat=display (and likely "explicit adult content" as well, although I haven't confirmed it). Note that anyone who changes this to "none" may not be able to login elsewhere with openID immediately after the change, there's some caching that will still prevent them from logging in to external sites for a bit afterwards.
I've also added this info to the original post.
If you still have problems after trying this fix (and the other fixes in the original post), please let me know :)