The turn with Recovery Disk didn't change a thing. I then installed a new User, no difference either, so my conclusion was, it had to do with the System. I opened the ~/Library there and dit some cleaning which finally worked for my system:
I went through all entries in ~/Library (the Library folder on top level) and cleaned some very old content form all the folders found there (some were 3-5 years old!). They must have gotten copied from older machines when migrating. I didn't touch the folders in ~/System/Library!
I opened each folder in list view, clicked the top to organize by date and then removed all files that seemed too old for this installation. You have to be careful, not to delete files belonging to old apps still in use.
In this way I could remove the misbehaviour without knowing which file was actually responsible. If I had deleted an old dated file of an app still in use, I could just have reinstalled it, but this was not the case. And removing one by one was just too much of research work.
I hope, someone can use these instructions for his or her own machine. Good luck.
Here is a list of the folders found in my setting:
Address Book Plug-Ins
Application Enhancers
Application Support
Audio
Automator
Bundles
Caches
CFMSupport
ColorPickers
ColorSync
Components
Contextual Menu Items
CoreMediaIO
Desktop Pictures
Dictionaries
DirectoryServices
Documentation
DropboxHelperTools
Extensions
Filesystems
Fonts
Fonts Disabled
Frameworks
Google
Graphics
Image Capture
Input Methods
Internet Plug-Ins
iTunes
Java
Keyboard Layouts
Keychains
Compositions
LaunchAgents
LaunchDaemons
Logs
Messages
Modem Scripts
Mozilla
PDF Services
Perl
PreferencePanes
Preferences
Printers
PrivilegedHelperTools
Python
QuickLook
QuickTime
Receipts
Ruby
Sandbox
Screen Savers
ScriptingAdditions
Scripts
Security
Services
Speech
Spelling
Spotlight
StartupItems
SystemProfiler
Updates
User Pictures
Video
WebServer
Widgets