For reason that neither I now nor Blizzard support (or they maybe did but did not share) or I found out from one day to the other I could no longer log into the forum only the main D3 web site.
After several updates to the original ticket and several emails received and “using” four different game masters on the “way”, I can finally log into the forum again. The “way” was a bit “painful”, but as I know from my former job - which included customer support as a small side job - there is sometimes no other “way” but the “painful” one.
What works is the solution provided below by the 4th of the game masters:
So if you are unable to remain logged in the forum by following the steps you showed in your picture, please try the following:
- Go to the forum
- Click on any post
- Scroll to the bottom of the post
- Click on the “Sign Up” button next to the “Maybe Later” button
- Login with your credentialsWith that you should be able to remain logged in the forums and post or interact with other posts.
The only step that I really could not do was
- Click on the “Sign Up” button next to the “Maybe Later” button
because I have/had neither of these buttons. So I tried clicking the “Reply” button instead. And to my surprise 
that worked. Now I only hope I do not have to log into the forum like that every time from now on. (Will update this post if I have to.)
Has anyone ever had that problem in the past? Because I could not find something about it in the Blizzard or the Diablo 3 forum (or did not research “good enough”).
UPDATE 1
It still does not work the obviously intended way. Only the workaround with the “Reply” button lets me log into the forum. As a conclusion this means that something in the code handling loging into the forum using the “Login” button (or keeping the credentials when loging into the web site first) needs fixing.
UPDATE 2
When I click “Log out” in the forum I get a very unexpected 403 error:
Is that “helping” with the problem?
UPDATE 3
It is fixed! Works like nothing had happened… 