thanks for the hint. It did not solve my problem, but it gave me an idea of the cause.
I think, user "system" or "admin" was missing to have rw-access in the home dir, in detail:
The newly created user, although of being an admin, had no rights. He could not start mail and system preferences. So I looked at the access rights in his home directory and realized that "system" has read/write access.
The existing user with my access rights problem had no user "system".
I tried to add "system" but failed. So I added the user "administrators/admin" to the list of users having read/write access in the home directory.
This solved the problem.
Someone (maybe me) must have deleted it before.