Personally, I hate the term "climate change". I think it's meaningless, for just the reason you mentioned. But, it's the term that the authors used. (Probably because it was required by reviewers and editors at Nature, I suspect.)

Their point on confirming "climate change" was that areas that were previously ice covered are turning green, which does confirm warming in some regions, thus - "climate change". I couldn't disagree with that. Of course, by itself it doesn't confirm the source of the warming, and it doesn't confirm any sort of global phenomenon. But that's just basic logic. The authors didn't say any of that... at least not in the Abstract, which is the part I focused on.

The main points that I thought were important from the article were: (i) the confirmation of wide-spread greening that is caused by CO₂ fertilization; and (ii) The confirmation that this greening removes additional CO₂ from the atmosphere.