In the same way drawing a line between programmable and fixed-program computer is arbitrary and in fact the author spends much of the article dismissing Babbage's Analytical engine, which was a programmable computer, on the grounds it wasn't actually built; which is, of course, irrelevant, because a physical machine is not necessary for complete computation, as demonstrated by Ada Lovelace.
In any case, the early days of modern computing (1930's- 1940's) were replete with a plethora of computing machinery of various architectures many of which were called "computers", without them being necessarily programmable computers; for example, the "Atanassoff-Berry Computer" [1].
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[1] https://en.wikipedia.org/wiki/Atanasoff%E2%80%93Berry_comput...
The point is there is a fundamental difference between a computing device which performs a single algorithm and a computer which can perform programmed algorithms.
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.
The author then argues that this categorization is unhelpful as it obscures the line between a purpose made computing device and a general computing device. The arrival of such devices he places at the end of the 1930s with the Z1, which was not made to perform a specific algorithm, but which instead allowed for programmability. Very soon after rapid advances happened, which created the "computer" as a general purpose computing device.
The Antikythera mechanism is essentially irrelevant to the point the author makes.