It's worth noting that in the case of logical induction, there's a more fleshed-out story where the LI eventually has self-trust and can also come to believe probabilities produced by other LI processes. And, logical induction can come to trust outputs of other processes too. For LI, a "virtuous process" is basically one that satisfies the LI criterion, though of course it wouldn't switch to the new set of beliefs unless they were known products of a longer amount of thought, or had proven themselves superior in some other way.
The definition may not be principled, but there's something that feels a little bit right about it in context. There are various ways to "stay in the logical past" which seem similar in spirit to migueltorrescosta's remark, like calculating your opponent's exact behavior but refusing to look at certain aspects of it. The proposal, it seems, is to iterate already-iterated games by passing more limited information of some sort between the possibly-infinite sessions. (Both your and the opponent's memory gets limited.) But if we admit that Miguel's "iterated play without memory" is iterated play, well, memory could be imperfect in varied ways at every step, giving us a huge mess instead of well-defined games and sessions. But, that mess looks more like logical time at least.
Not having read the linked paper yet, the motivation for using iterated or meta-iterated play is basically to obtain a set of counterfactuals which will be relevant during real play. Depending on the game, it makes sense that this might be best accomplished by occasionally resetting the opponent's memory.
I think it's worth mentioning that part of the original appeal of the term (which made us initially wary) was the way it matches intuitively with the experience of signaling behavior. Here's the original motivating example. Imagine that you are in the Parfit's Hitchhiker scenario and Paul Ekman has already noticed that you're lying. What do you do? You try to get a second chance. But it won't be enough to simply re-state that you'll pay him. Even if he doesn't detect the lie this time around, you're the same person who had to lie only a moment ago. What changed? Well, you want to signal that what's changed is that some logical time has passed. A logically earlier version of you got a ride from a logically earlier Ekman but didn't pay. But then Ekman put effort into remembering the logical past and learning from it. A logically more recent version of you wasn't expecting this, and perished in the desert. Given that both you and Ekman know these things, what you need to do in order to survive is to show that you are in the logical future of those events, and learned from them. Not only that, but you also want to show that you won't change your mind during the ride back to civilization. There will be time to think during the car ride, and thinking can be a way of getting into the logical future. You want to demonstrate that you're fully in the logical future of the (chronologically yet-to-be-made) decision to pay.
This might be an easy problem if the hitchhiker and Ekman shared the same concept of logical time (and knew it). Then it would be similar to proving you remembered the literal past; you could describe a trivial detail or an agreed-upon signal. However, agents are not necessarily incentivized (or able) to use a shared imaginary iterated version of whatever game they're playing. To me it seems like one of the real questions the logical time terminology brings up is, when, and to what extent, will agents be incentivized to use compatible notions of logical time?
What does it look like to rotate and then renormalize?
There seem to be two answers. The first answer is that the highest probability event is the one farthest to the right. This event must be the entire Ω. All we do to renormalize is scale until this event is probability 1.
If we rotate until some probabilities are negative, and then renormalize in this way, the negative probabilities stay negative, but rescale.
The second way to renormalize is to choose a separating line, and use its normal vector as probability. This keeps probability positive. Then we find the highest probability event as before, and call this probability 1.
Trying to picture this, an obvious question is: can the highest probability event change when we rotate?