The full error is:
Message: Mysqli statement execute error : Can't create/write to file '/var/www/zendApp/backup/mysql/1268733580_name.sql' (Errcode: 13)
I have tried several things already with the /var/www/ZendApp/backup/mysql directory
$chmod 777 mysql
$chmod 1777 mysql
$chown root:root mysql
$chown mysql:mysql mysql
But nothing helps. What am I doing wrong here? Do I have to tell mysql somewhere that I can write to that directory?
With the help off Dominik @ Super User, I discovered that Ubuntu is running apparmor and my server is running on Ubuntu. I quote from wikipedia:
AppArmor allows the system administrator to associate with each program a security profile that restricts the capabilities of that program
Sure enough, among those programs was mysqld. So not the user or his privileges are the problem, but the restrictions imposed on the process by apparmor. I then found the solution here
sudo touch /etc/apparmod.d/disable/usr.sbin.mysqld
sudo /etc/init.d/apparmor reload
This makes mysqld a free process which for me, on that server, is no problem. So, after days of searching, problem fixed!