My reading:

The article is saying that neither "virtual particles" nor "particles" are good descriptions of reality. Those are better understood as just names for different shapes of ripples in a field. In the field picture, there are no particles as you would normally think of particles, but there are ripples which a variety of behaviours.

In the article's suggested model, "particles" are ripples that travel away from whatever caused them and carry on. They retain structure that they carry with them, so as they move away from the process that caused them, we think of them as objects in their own right.

"Virtual particles" are ripples, but they don't propagate away. Think of the difference between evanescent and propagating waves. Evanescent waves are ripples or shapes in a medium that don't propagate. Virtual particles are like these. There's still a ripple, a bit like the wake near a boat, but the shape of the ripple means it doesn't propagate away from whatever caused it as if it was an object in its own right.

"Virtual particles" is unhelpful terminology, because they don't behave or even exist the way we imagine a particle. Descriptions of virtual particles popping in and out of existence are misleading, as they are not particles, don't behave like particles, and there are no sudden events. There's no popping.

In Hawking radiation, when they say a pair of virtual particles appears, it means certain non-propagating ripples form in the field. That part happens normally without a black hole as well. It means there's a local ripple in the field that is non-propagating (like evanescent waves).

But in Hawking radiation, the interaction between these quantum field ripples and the distorted geometry of spacetime caused by black hole gravity means part of that local ripple ends up propagating away. The geometry of spacetime turns what is normally a local, non-propagating ripple into a propagating ripple.

In the language of "virtual particles", we wouldn't say the virtual particle propagates away. Rather, we'd say a process takes place where a virtual particle interacts with event-horizon gravity to create a (non-virtual) particle.

But it's more accurate to stop thinking about particles, and say what is normally a non-propagating ripple becomes propagating in the presence of spacetime geometry distortion, and that carries away energy. All these processes and geometric distortions still conserve energy, so any field configuration which produces a propagating ripple away from the black hole also results in the black hole itself losing a little energy, and therefore losing a little mass (because energy and mass are equivalent), which is why Hawking radiation causes black holes to shrink.

Black holes and event horizons are pretty screwy anyway, as the very notion of whether something exists or not, and whether something happens or not, can depend on your frame of reference.

> Or is the article calling a "disturbance" the creation and annihilation of a pair of virtual particles? I certainly see how that's different from a particle, but it's also pretty clear that it's not one particle but a pair. Of course this very special pair of particles that spawns and disappears is not like "a" particle.

It's the other way around: When a text (or mathematical abstraction) says creation and annihilation of virtual particles, it's misleading picture because virtual particles don't exist, and they don't pop into and out of existence. The compound operation you might call "creation-and-annihilation" is a way of describing a non-propagating local ripple, which has some mathematical subcomponents that it's convenient to call creation and annihilation of virtual particles, but the parts always co-exist in any event. There are no separate creation and annihilation events of these; they always co-exist.

Then in Hawking radiation that picture isn't quite right. One way to describe it in virtual particle language is add gravitons to the picture, but that might be a hack. Another way is to drop the virtual particle language altogether, as they are a mathematical tool and nobody has found a way to make them work consistently with gravity anyway. Even looking directly at the behaviour of fields might not give the right answers: Famously QM and GR have yet to be reconciled.