While funny, I don't think this has much bearing on the topic at hand: we share a lot of DNA with chimps as well (or rather, the last common ancestor between chimps and humans). But that doesn't mean we should call chimps human.
So the shared genes alone can't be used as a reason to call Neanderthals human.
No human had a chimp ancestor. But humans share a fairly recent common ancestor with chimps.
Some human had Neanderthal ancestors, some did not, as you rightly suggest. However, all humans and Neanderthals share a common ancestor that's much more recent than our last common ancestor with chimps.
Ie we only have 'chimp DNA' in the sense that we share a lot of DNA with them, and that we haven't changed too much since our last common ancestor. Exactly the same is true of us and Neanderthals.
A single-word common name usually refers to a genus: "oak" is anything in Quercus, "wagtail" is anything in Motacilla and so on. That's not conclusive because there are plenty of exceptions, and "human" could easily qualify as a special case, but I don't see why "human" shouldn't be any member of Homo.
