By "artificial light", they mean tungsten (incandescent) and fluorescent. Flash is artificial too, but it's usually relatively high color temperatures (like the midday sun) and therefore not hard for the auto-wb systems to get right.
In any case, it is a moderate concern. It's not necessarily a deal-breaker because there are plenty of workarounds, but even if you shoot raw and post-process every image, it's nice for the white balance to be at least close to correct initially.
My current camera, a Pentax K-7, has incredibly good automatic white balance, usually providing results better than the presets like "sunny" or "cloudy". I find this to be a huge benefit and time saver. I'd be disappointed with any camera that had worse performance in this area. (The K-7 has a sensor that it uses to adjust the autofocus system depending on the wavelength of light, and it's speculated that this is also used for the auto-wb, but I don't know of any proof of that and Pentax won't say.)
Unfortunately, online reviews don't seem to cover this very well, either accepting poor performance as the status quo and maybe putting a line like the one you noticed in the conclusion, or else using some sort of technical measuring system where the results don't match with my experience. Not to argue with science, but there's a lot of ways to get those technical measurements wrong, and since they don't line up with my actual observations, I'm skeptical. So this is something I'd test out in person first if possible, and ask someone who has the camera about if not.