There's no standard definition for the term "multiverse", because it's not a term that corresponds to any established physical theory. The theory described in the article has a good claim to the term multiverse, because it's much more than just separate regions.

The region of the universe we're in almost certainly extends beyond the limits that we observe, so there are already "separated observable regions" in the universe we know. The article is talking about a scenario in which multiple Big Bangs occur, so each region is not just separated by distance but also by the nature of the space in that region - how much it has inflated, how fast it is inflating. Each such inflating region is possibly also distinguished by different laws of physics in that region. There would also be non-inflating regions which would have properties different from anything we're familiar with.

Back when other galaxies were first discovered, they were originally referred to as "island universes". This eventually changed to "galaxy" as our understanding of the extent of the universe shifted. If the theory in the article were somehow confirmed (difficult!) then in future, we might indeed refer to that larger space as just "the Universe", and refer to the inflating bubble we're in as something less all-encompassing than "the Universe". For now, though, it would be very confusing if we started referring to speculative constructs way beyond our ability to observe as "the Universe". Multiverse is as good as a term as any.

Sounds like the blind leading the blind.

You can't naively apply Popper in this case (who in any case is by no means the last word on philosophy of science), because the situation is quite complex: the article describes a possible consequence of existing established theories, including quantum field theory, general relativity, and Big Bang cosmological models. As such, Popper's rules don't say anything about those theories not being science, or whatever.

While it's true that "the math does not lead only and exclusively to that conclusion", it's a valid possible conclusion. As such, given the status of the theories that it's based on, we can't avoid taking it seriously as a possible description of reality. The task then becomes to discover if there's any way to improve our certainty about its correctness or lack thereof, and that's why people like Linde write papers about this stuff. Rejecting this as "not science" or whatever based on one particular view of what science is, is terribly short-sighted, and it's lucky that actual scientists don't pay attention to such nonsense.

One of the interesting consequences of eternal inflation style theories is that in principle, it addresses questions of fine-tuning. One can take the "evidence of fine tuning" as an argument in favor of multiverses in some form. From that perspective, the idea that our observable universe that started with the Big Bang is the only universe is actually the more difficult theory to defend, since we don't know how some of the parameters managed to come out on the knife-edge of allowing the universe to expand to a useful size and have useful properties like the ability for matter to form.

Re Popper, you should look into Imre Lakatos, who pointed out various flaws with basing all of science on falsificationism. See e.g. The Methodology of Scientific Research Programmes.

No cosmologist says that. The edge you're referring to is the edge of the observable universe, nothing more.

FTFY