Does creatine-phosphate (CP) supplementation regenerate NADH?
Creatine itself is never converted into ATP. Creatine-phosphate on the other hand can donate its phosphate group to ADP, phosphorylating to form ATP and creatine. This is a buffer system for high-energy phosphates, and is very important in organs with rapid ATP turnover, notably muscle.

The mechanism of the findings described in the papers is not fully understood yet I think. But it is not via ATP, because the supplementation is not creatine-phosphate, but creatine itself, which does not yield ATP.

Regarding NADH: ATP is very different from NAD/NADH in terms of molecular structure, and the two are not interconverted by any pathway that I know of. (Probably this is not what you meant, but just to be clear.) But it is certainly true that reducing power in the form of NADH can be used to phosphorylate ADP into ATP; this is the task of the mitochondrial respiratory chain, where NADH is oxidized to NAD by complex I. On the other hand, I am not aware of a physiologically relevant process that uses ATP to reduce NAD into NADH.

The liver is a main site of synthesis of creatine itself, and this syntesis consumes methyl groups donated by the S-adenosylmethionine system. The authors of your ref1 seem to suggest that reduced synthesis of creatine following supplementation is a possible mechanism. Similar effects have been observed with choline supplementation, but again this effect is not well understood and a current research topic. A recent review is found here.