What a fascinating discussion!
IMHO, there are two concepts that seem to easily confuse our understanding of sati:
1. Remembrance: we often equate remembrance to thinking. But it is not. Actually, remembrance is just a mental factor which arises and performs the function of remember, together with a citta knowing an object at that every moment. Sati is also a mental factor, if we consider that way, then it is easier to make the connection between the two.
To understand the remembering aspect of sati, just think about the moment between a state of forgetfulness to a moment of awareness. What the mind knows at that very moment? It knows it has been forgetful, and also what has been forgotten - that is remembrance -sati at work.
2. Present moment: What is the present moment? Time is created by the mind.
For practitioners, we feel that we are in the present moment when there is no thinking of past or future, or where we are aware of what is occuring in our body and mind, right?
But are we actually aware of every dhamma that arises and passes away? There are billions of them in each second. Are we sure we are aware of the moment of arising to the moment of passing away of each dhamma in its minute existence?
With practice, we know that the more we are "in the present moment", the more we feel like catching a fish in the water with only one hand, as phenomena arise and pass away extremely fast.
That means "being in the present moment"- so to speak- doesn't negate the remembering aspect of sati, because when sati remembers two or three or hundreds previous dhammas, we are still perfectly in the "present moment".
I believe that it is where sampajana becomes stronger that it see better the working of sati in details.
The first vipassana nana consists of understanding nama and rupa separately, as dhammas, no person , that means sampajana is strong enough to see sati remembering the "dhammas" that have arised.
My understanding is that, from the vipassana stages, sati goes further than remember learnt right view, to directly remember "dhammas", establishing thus a superior kind of right view.