We think of a lightbulb as a device which generates light, but it isn't that intrinsically; the filament glows only when electrified, and the bulb is only a small part of the light-generating system which includes an electrical generator (and, presumably, a burning fuel source). The bulb is the place where the light first becomes visible, but it makes less sense to say that it is the place the light comes from.
Similarly, we think of the brain as a thinking machine, but it isn't that intrinsically; it is only a small part of a much larger thinking system which includes functions of respiration, circulation, ingestion, et cetera. So we can say that the brain may be the place where thinking first becomes noticeable per se, but it makes less sense to say that it is where thought is created.
Which is all well and good. I'm not sure what part of it is supposed to be a hard question about neurology rather than a PHIL100 essay topic, though.
How to distinguish between the two might be an interesting question from, say, a Chinese Room[1] or p-zombie[2] standpoint, which might have some philosophical bearing on AI. But that's about all I can think of off the top of my head.
As far as the original article, I think anyone who tries to argue that you are not your brain is ignoring over a hundred years of scientific evidence that suggests otherwise.