In general, that is what makes half of foreign policy reporting semi-useless, newspapers rarely report in which sense diplomats use the words. (Compare "fighting terrorist" as claimed by everybody, in general terrorist is just defined by threatening the interests of the speaker.)
[1] Likely meaning a mutual, synchronous and long term reduction of nuclear weapons towards a global zero goal. (Like the stated US nuclear policy of the Obama administration.)
[2] Meaning a prompt and unilateral abolishment of the NK nuclear program, as in Libya.
There are so many worthy recipients of the Nobel Peace Prize that are virtually unknown outside their specific non-profit niches — focus on those instead, and use the prize to highlight their work.
Still, judged by the outcome, it's kind of impressive. ;-)
He first called him Rocket Man, which was an ingenious way of telling him "OK, you have achieved what you wanted" and praising him in a way. He pulled Kim into the eye of the public in a non-humiliating way (the "short and fat" was a response to Kim calling him senile first).