I have no idea what was going on with the Ministries Tribunal and cannot speak to what they were thinking when they applied different standards -- assuming that it is true that they did.
But the other questions are easier to answer:
Purpose vs. knowledge:
The distinction between purpose and knowledge is a tricky one, but some hypotheticals can help tease out the difference, which centers on what the actor is truly trying to do:
One night, a bomb goes off in a synagogue, killing a cleaning woman inside. The next night, another bomb bomb goes off in another synagogue, killing a cleaning woman inside. The local Jewish population is terrified of what might come next. The police arrest Man 1 for the first bombing and Man 2 for the second. They charge both with:
- terrorism, which we'll define as "the purposeful use of an explosive to intimidate a religious group."
- murder, which we'll define as "the purposeful killing of another person."
Man 1 is the husband of the cleaning lady at the first synagogue. He is Jewish and generally harbors no ill will toward other Jews. But he's a drunk and a loser, so his wife decided to leave him. Humiliated, he decided to kill her by rigging a bomb to go off in the broom closet when she opened it. As he did it, he worried about his congregation thinking they were being attacked because they were Jewish. He knew that was what they would think, but he was too angry at his wife to care.
Man 2 is a Nazi. He hates the Jews and he bombed the synagogue because he wants to spark a race war. After installing the bomb, he went outside to detonate it. He knew a cleaning woman was inside. He knew she wasn't Jewish and did not want to kill her. But he figured that sparking a race war is going to mean some innocent people get killed. He felt bad that she would die, but he pressed the button anyway.
The men's guilt or innocence will depend on why they put the bombs in the synagogues.
Man 1 can be convicted of murder. He knew his bomb would kill someone, and he put it there for that purpose. He cannot be convicted of terrorism. He knew that local Jews would feel intimidated by the bomb, but his purpose was not to intimidate anyone.
Man 2 can be convicted of terrorism. He knew his bomb would intimidate the local Jewish population, and he put it there for that purpose. He cannot be convicted of murder. He knew that his bomb was going to kill the cleaning woman, but his purpose was not to kill her.
Another common hypothetical is the woman who poison's her husband's coffee thermos. She wants to kill him, but she knows that he always shares his coffee with another guy at work each morning. She doesn't want to kill him, but she does it anyway. She has acted knowingly and purposefully in killing her husband. She has only acted knowingly in killing husband's co-worker.
Alien Tort Statute: The purposefulness standard was considered a "death knell" for the Alien Tort Statute because it is so much harder to prove than a knowledge standard. If someone fires a gun into a crowd, there's always going to be a question about what the purpose was, but it's a lot harder to prove that they didn't know it would kill someone.
The same is true with the Alien Tort Statute. Presbyterian Church of Sudan v. Talisman Energy, 582 F.3d 244 (2d Cir. 2009), dealt with claims that Talisman had facilitated Sudanese war crimes. Those claims stemmed allegations that Talisman had upgraded Sudanese air facilities, provided financial support to the Sudanese government, and provided logistical support to the Sudanese military, all while Sudan was allegedly committing war crimes during the Second Sudanese Civil War.
While it would be relatively straightforward to prove whether Talisman knew about the government's conduct in prosecuting that war, it could be more difficult to prove why they took the actions listed above. If the ATS imposed a knowledge standard, it would not matter why they provided that assistance. But if it imposed a purposefulness standard, they would have a relatively straightforward defense by claiming that they were doing so only to protect their investments in the oil fields around the country. And if that's the case, it would matter even if they knew for certain that making those investments would lead to genocide, war crimes, or other crimes against humanity.