Since I have not received any further suggestion on the issue, I have to assume there no longer any available.
This means, I have to bite the sour apple and do a complete reinstallation of Windows 8, Windows 8.1 Upgrade, and then reinstall all applications I had. This shall then resolve the issue.
What I cannot understand how this could happen, because the system runs otherwise fine, and the Windows Update ran also fine until June 9. Secondly, I ran a thorough virus scan several times and this did not turn up anything pointing into this direction that there is something wrong. From all of this as I mentioned before, the only conclusion I can make is that the Windows Update component is massively corrupt or something is totally missing. It is difficult to say what the latter could be.
With my action to reinstall, I try to avoid running into another wall when Windows 10 comes up, and it then tells me it cannot upgrade because of the issue I am facing now.