So a few uses of "arising" are being used in Ven. Nanananda's passage:
samudaya: rise, origin, produce
uppāda: an unusual or startling event, coming into being, appearance; production
In the below sutta (which I know I quote a lot) we see uppāda and also paññāyittha, which means "to be perceived, seen or taken for, to appear".
We would first have to agree that these three terms are hovering around the same idea, and that since we see these terms all associated with descriptions of the aggregates, in the very least they can all be applied to the nature of the aggregates in some way. Maybe not in the exact same way, but they are close.
As the sutta goes on to say, and as Ven. Nanananda points out in the passage, the answer is the five aggregates, but if we back up and look at the first line, "...arising is manifest, ceasing is...", it is another way of saying an arising has appeared, ceasing has appeared.... That "manifestation" seems to be a more general way of categorizing the appearance of these different natures, distinct from the natures themselves. So it would seem that appearance is what arises, and is a more general view than the particular aggregate in question, i.e. to appear is the nature of the aggregates and whether or not they are doing so as arising, ceasing or persisting-while-changing does not alter that most general nature which is to manifest.SN 22.38" wrote:"What are these things of which and arising is manifest, a ceasing is manifest, a change while standing is manifest?"
This surely complicates the common notion that experience is a simple arising and ceasing of things all the time. The ability to observe that nature, i.e. seeing would have to be enduring throughout the arising and ceasing in order to call a spade a spade. So there is more to it than just those two movements.
With that, and to answer your question, I'll put it this way (imo of course): the appearance of the aggregates in these different states is what arises. They are not just coming and going, but indeed they are changing, or rather they are - according to SN 22.38 - growing in one sense, decaying in another and in yet another they are enduring . They are "heaps" and they accumulate layers (even layers that mark a deterioration of that accumulation). But most importantly (and it all that really matters), they arise in a state of being clung to (upādānakkhandha), they arise as mine, belonging to me ---- already in that state. In satipaṭṭhāna, there is a heavy emphasis on the presence of "I am doing..." in every contemplation. And if you look at MN 10 for instance, through every mindfulness contemplation: in general it starts out with "I am doing" and then comes to a close with something along the lines of "There is body" "There is feeling". The repeated mindfulness has cleared that present "I" and separated it out of the picture where it once dominated. (cf. Ven. N. Nanamoli, Notes on Meditation).
So I see no reason why vipassanā would not just follow suit, considering how it extends from satipatthana. The goal is to find that separation between experience with "I am" and experience without it. And the condition that the aggregates are in is critical to that insight.
Jeez I hope a came in clear, can't tell at this point. Too tired.