I think the claims in the piece are fairly robust to changes in the definition of economic growth. That is: for
any reasonable definition of terms like "economy", "wealth", etc., it seems (1) fairly clear that "the economy", meaning in some sense the total amount of actual "wealth", has been growing at a few percent per year on average for quite a while, and (2) monstrously unlikely that "the economy", meaning in some sense the total amount of actual "wealth", can continue to grow at a few percent per year on average for, say, the next 10k years.
(Because, as the author says, that would mean having as much of whatever-it-is-that-economies-are-made-of per atom in the galaxy as we currently have in total.)
So for that sort of growth to continue for that long, something very radical would have to change. (We discover how to teleport arbitrary distances, or to move to entirely different universes, so we are no longer limited by the size of this galaxy. It turns out that humans really are immaterial souls that just live in material bodies, and we find a way to break free of the physical world altogether. At least that sort of level of radical-ness.)
Hence, either the growth we are all used to will have to slow dramatically, or some other radical change will have to happen, or both.
None of this depends on the details of how you define "the size of the economy" -- though, as you say, some particular choices you can make (but that no one ever actually does) would invalidate the argument.
(When people talk about economic growth, they never mean growth in nominal dollars or whatever. That would be silly. They never mean mere population increase, independent of the wealth of that population. That would be silly.)