Answers to #1:
Raphael Lemkin’s definition of genocide was not accepted until after the Holocaust.
Raphael Lemkin had been studying the problem of mass killings of a people group since the 1920s, in regard to Turkish slaughter of Armenians in 1915. He coined the term “genocide” in 1944, in reference then also to the Holocaust. The term uses Greek language roots and means “killing of a race” of people. Lemkin served as an advisor to Justice Robert Jackson, the lead prosecutor at the Nuremberg trials. “Crimes against humanity” was the charge used at the Nuremberg trials, since no international legal definition of “genocide” had yet been accepted. Ultimately, Lemkin was able to persuade the United Nations to accept the definition of genocide and codify it into international law. In December, 1948, the United Nations adopted the Convention on the Prevention and Punishment of the Crime of Genocide, which made use of a number of Lemkin’s ideas on the subject.
#2: For item #2, you didn’t ask a question, so I won’t attempt to guess at what question you might have in mind. The definition as you quote it comes from Article II of the UN’s Genocide Convention. Article III also indicts intention and conspiracy to commit genocide as crimes against international law. Article IV of that same Convention then puts teeth into the UN’s action, saying, “Persons committing genocide or any of the other acts enumerated in article III shall be punished, whether they are constitutionally responsible rulers, public officials or private individuals.”