How about a device that can perform more than one computation then? e.g. the Antikythera mechanism could calculate the motion of the planets, and, separately, eclipses. Universality is the only characteristic that makes a difference, but I think the author avoids this because they don't want to be caught up in debates about whether modern computers are truly universal.

The article is trying to make the distinction between what is, and isn't a "computer" sound crisp and exact, when it's arbitrary and ad-hoc.

Yes, more or less. To quote another commenter in another thread, " an open definition of what a computer is, is too inclusive and uninteresting if you want to talk about the history of actual computers".