So you dont have a source of this definition? Because in international law its defined quite differently, which is quite obvious since your definition, as given, would not classify many real instances as genocide.
It’s the entomology of the word.
Legally they have a definition which broadens the concept to not just direct killing of a people, but also acts that lead to the eradication of a culture. For example: kidnapping the children of a group and raising the within your own culture is a genocidal act as it leads to the culture dying off, even though no actual killing has taken place.