The problem here (and I caught it too and sort of recoiled at it) was they were using their logic and worse language/word choice regarding the "abstract machine

The problem here (and I caught it too and sort of recoiled at it) was they were using their logic and worse language/word choice regarding the "abstract machine" being the actual thing we should think about when reasoning about a programming language before they actually went out and were explicit that's what they were doing. They saved the explicit thesis until the end of the piece. They should have traded the writer's desire to be witty and hide the lede and just lead with it upfront. Of course you're right in part that their idea of "memory" is an abstraction, but it isn't too wrong. A "variable" in C or any compiled language on a modern machine is an abstraction that could refer to a register one moment, be a place in the cache in another, be a place in memory in the next, and be on a swapfile after. The "variables" are abstractions which lie in "memory" which is another abstraction because it need not be in one place.