Did Minoru die because of the Death Notes' new rule or because Ryuk wrote his name in a Death Note?
Minoru died because of Ryuk, similar to how Yagami Light did.

Through the original Death Note Series Ryuk mentions that when the owners life would come to an end, he would write down the name instead. However, Ryuk has also shown that he is impatient in that sense.

As Minoru has broken the new rule, and obtained his payment Ryuk wrote down his name, as Minoru was set to die anyway.

The One Shot however does not cover whether this new rule would actually kill Minoru off even without a Shinigami writing his name down. However, knowing how Shinigami themselves have also died from breaking the rules set upon them, there is a high likelihood this would have happened non the less. Where I would speculate that the Shinigami King himself would be the one to pass down the punishment for breaking the rules.


  You have lost, Light. Didn't I say in the beginning… when you die, the one who'll write your name down in a notebook will be me. That is… the deal between the Shinigami… and the first human to get their hands on the note in the human world. Once you enter the prison, I don't know when you'll die. It's annoying to wait… Your life is already over. You'll die here. Well, it was good while it lasted… We killed some boredom, didn't we? We did some various and interesting things…” - episode 37