To some extent, I think, the division in the ranks is due to prior commitments, whether to apocatastasis (and “restorationism” is a very good translation of that term) or to eternal punishment. This can be seen clearly in some earlier studies of the two terms, which we refer to in our book: in these, the motivation was explicit. It’s why I insisted that we avoid theological issues and focus on the words themselves.
Now, the words are not always perfectly clear in their meaning, and sometimes the sense appears to depend on the context. Still, there is no question but that the early sense of aion is simply a lifetime, and that it comes to mean also a long period of time, an eon or age; and aionios follows suit. Plato introduces the confusion, but employing aionios for his new idea of a timeless eternity, since aidios was too clearly connected with a time extending to infinity. Still, in classical literature the difference is fairly clear.
Then, with the Septuagint, aionios becomes the common term, and so too in the NT. It’s meaning here depends in part on the sense of the Hebrew terms that lie behind this usage, but since koine Greek is subtly different from classical, one has to do the work separately on these texts.
Now, aionios is most certainly applied to God, and my colleague Ilaria wants it to mean infinite in that case; of course, God is infinite in every respect, but is this the force of aionios when so used? My sense is that it still bears the connotation of belonging to another aion or epoch, and can mean something like “transcendent,” but it’s a delicate issue. At all events, here is where there is space for disagreement.
We did our best to lay out the evidence fairly, and came down on the side of restorationism.
I hope this helps,
Very best, David