I found it out myself... In case someone is interested:
1) install your stylesheet (see above)
2) Format/Edit Stylesheet
3) Click on your Stylesheet notebook file
4) Option Inspector/Notebook Options/File Options
5) Set Saveable on True
Now you can save your stylesheet notebook with File/Save after modifying styles.
Remains the question why this is not the default setting...
Hey, I am back. I hoped that with version 10 of Mathematica managing styles would became a bit easier. However, it seems that it's still the same problem: I still don't know how I can directly save a changed style within a stylesheet without passing by "save as" and then reinstalling it. I change styles all the time so this is really a time consuming method. What is the reason I cannot simply save my personal stylesheet using File/Save? Is it not saveable because it inherits it from default.nb? Or is it just a bug? How can I make it saveable?