I don't think the writer is really doing anything much wrong. In the Wikipedia article on the topic "convex hull" is defined as you suggest for sets in abstract, but for a finite set of points, as being the polygon (and not the interior of the polygon), which is consistent with the writer's terms and it is rather closer to both the intuitive understanding of the matter at hand and the algorithms under discussion.

It seems to me that the term "convex hull" is ambiguous, even among geometers -- sometimes meaning the set and sometimes the boundary. A hull tends to be hollow, so I'd suggest that the boundary definition is nearer normal usage and intuition. In any event, the writer defines the terms and uses them as defined.

I think the two algorithms he examines are interesting. The second is the one I first think of. The two methods are almost duals of each other. It seems to me that if you start by taking the top, left, right, and bottom points as your initial hull and then add on for any points not in that hull, it's almost certainly faster than either algorithm (especially in practice).