If I write: John Zeürtö kills himself by writing his name in the Death Note. Would he do so? (In this example, John also owns a Death Note)
Can you manipulate someone with the Death Note, so that the person writes his own name in the Death Note?
Yes, you could and that person will do so if he is physically capable of writing a name in a Death Note, but it wouldn't really matter what kind of notebook it would be. Nothing would really happen when he writes his own name, because of How to Use: XV
When the same name is written on more than two Death Notes, the note which was first filled in will take effect, regardless of the time of death.
Can you manipulate John Zeürtö to kill himself by writing his name in the Death Note, by using a Death Note?
No you can't. If you do so, his name would already have been written in the Death Note, so any further name writing, would have no effect, as mentioned earlier. Therefore he wouldn't be able to kill himself. This would lead to How to Use: VI and he would die of a heart attack, since no other cause was specified.
The conditions for death will not be realized unless it is physically possible for that human or it is reasonably assumed to be carried out by that human.