Difference between abiding nirvana and non-abiding nirvana?

"Omniscience has regularly been ascribed to the Buddha in the different Buddhist traditions. An examination of the early discourses found in the Påli Nikåyas and the Chinese Ógamas, however, suggests a different perspective. The term used in the Påli Nikåyas to qualify someone as omniscient is sabbaññu, with its counterpart in the Chinese Ógamas in the expression 一切知, (yi qie zhi). The term sabbaññu and its equivalent 一切知 are made up of two parts: sabba or 一切, “all”, and ñåˆa or 知, “knowledge”, just as the English term omniscience derives from the Latin words omnis, “all”, and scientia, “knowledge”. In the thought world of the early discourses, such omniscience denotes the ability to continuously and uninterruptedly have complete and infinite knowledge regarding any event, such as is attributed by theistic religions to their god(s). An instance where such infinite and total knowledge is attributed to the god Mahåbrahmå can be found in the Påli and Chinese versions of the Kevaddha Sutta, according to which other gods in the Brahmå realm believed that there is nothing Mahåbrahmå does not see or know. The same discourse also depicts the early Buddhist attitude to such claims, as the Påli and Chinese versions agree in describing how Mahåbrahmå reacted when faced with a question posed by an inquisitive monk. In order to avoid losing face in front of the other gods, Mahåbrahmå gave evasive replies, until finally he had to admit that he did not know an answer, and that the question should better be put to the Buddha, who would be able to solve it." :from: "The Buddha and Omniscience by Anålayo∗ - https://www.buddhismuskunde.uni-hamburg.de/pdf/5-personen/analayo/buddha-omniscience.pdf

This is an interesting problem, but important because it helps us to understand why in some cases, enlightened teachers are known for unenlightened views. They themselves are free, but they don't have something akin to omniscient knowledge (I mean that in a relative way) as Buddhas do. Buddhas teach Dharma from full recollection of the entirety of incarnations, so every little detail of knowledge has been worked through in their past. So for example, a Zen Master in Japan might has realized enlightenment, but being ignorant due to his past of only seeing things in a Japanese way, might voice his support for seeing the Emperor as Universal or World Monarch for all of Asia.

"Victoria has followed up his Zen at War with more research. He has directed attention to Yasutani's Zen Master Dogen and the Shushogi (Treatise on Practice and Enlightenment), published in 1943. This book is "a rallying cry for the unity of Asia under Japanese hegemony": Annihilating the treachery of the United States and Britain and establishing the Greater East Asia Co-prosperity Sphere is the only way to save the one billion people of Asia so that they can, with peace of mind, proceed on their respective paths. Furthermore, it is only natural that this will contribute to the construction of a new world order, exorcising evil spirits from the world and leading to the realization of eternal peace and happiness for all humanity. I believe this is truly the critically important mission to be accomplished by our great Japanese empire. In fact, it must be said that in accomplishing this very important national mission the most important and fundamental factor is the power of spiritual culture." -from: https://en.wikipedia.org/wiki/Hakuun_Yasutani

I've always assumed that abiding Nirvana is, well, abiding, in that we are fully aware and awake and nothing can change this. Non-abiding Nirvana would be a glimpse or a taste but not a settled state. The former would be the goal.

This view is not shared by all, I note, so it's just my opinion.

This is a very interesting excerpt. Unfortunately I can not answer your question about the Sanskrit terms, but I hope what I have written is still helpful to your understanding of the difference between these two states.

Once Nirvana is reached, as described above, there is an instantaneous liberation of the conception of existence, and true (or I would call "fundamental") suffering. That being said, although Nirvana is known, there are still habits and experiences that arise based off ignorance. This is because the habits of the illusory self continue to appear until the Truth of Nirvana becomes incorporated into the experience of the illusory self.

This state of being awakened, where the illusory self that is appearing has not fully digested and processed the understandings Nirvana brings, is what they are describing as "abiding nirvana". It is called abiding, because it feels as if the default existence is still the illusory self, and it takes a conscious effort to "abide" in Nirvana. This ability is there at any time because you are enlightened.

Now, non-abiding Nirvana, is the state that occurs after all ignorance and habit is removed from the illusory self's behaviors and understandings. In this state, the default existence is no longer the illusory self and it no longer feels like you need to put effort into abiding in Nirvana. From here, the default perspective of your existence is one that is Nirvana first, and all ignorance has been fully dissipated through true understanding.

An Arahant who is in the "abiding" stage is capable of feeling things such as anger, frustration, and desire. Although these things only arise out of habit, and when they do arise, they quickly are dissipated for the Truth is known.

An Arahant who is in the "non-abiding" stage no longer observes the arising of things like anger, frustration, and desire. This is because the illusory self has now fully embodied the teachings of the Buddha. Once this has occurred, there is nothing left to be done.

The statement "his enlightened activities are uninterrupted" are referring to the wavering back and forth between illusion and Nirvana. When enlightened activities are uninterrupted, you are continuously resting as Nirvana, and no content that appears within the story of the illusory self is capable of pulling your attention back into "embodying" the illusory self. This state of non-abiding is a gradual transition that occurs over time, as the mind processes the information it can no longer deny after observing the Truth.

I hope these definitions where clear and helpful in your understanding of the difference. I wish you nothing but love and pray you reach liberation soon.