If one wants to get nitpicky, perfectly normal warfare already falls within the defintions of the bit you quoted... since two warring nations' armies are by default "killing members of the group; causing serious bodily or mental harm to members of the group; deliberately inflicting on the group conditions of life, calculated to bring about its physical destruction in whole or in part" and so on. By that same token the Lusitanians were engaging in "genocide" every time they bushwhacked a Roman army, and the Cisalpine Gauls every time they raided into Italy proper... heck the Celts and Germans and all the others with traditions of raiding their neighbours would have been engaging in "genocide" against themselves, Alexander the Great against the Greeks when he had Theba razed, and let's not even get started on everyone involved in the Thirty Years' War...

I don't particularly like this way the concept of "genocide" tends to get banalized these days. It dilutes the weight and meaning of the term when it is applied to simple bloodbaths and massacres. IMO it only becomes valid for those when they are part of an actual campaign to genuinely exterminate a people - but hardly when the ultimate goal is defeating and subjugating the other.