If it were between two dslrs, I would have no clue, but I know the Fuji has a different kind of sensor and a different look, so I'll give it a shot. First set is Fuji, based mostly on how the white areas look.
You are definitely right!
I do not know much difference between other DSLRs. But I noticed the feeling of the photos from two cameras is different. I think I prefer the second set (D810), which I think benefit from higher pixels and bigger sensor (which caused shallow depth of field)
My boyfriend had a Pentax K5 and then got a Fuji X-something (one with interchangeable lenses). He did a few comparisons by taking the same picture at the same time with the two cameras and we looked at them side by side. As good as the Pentax images are (and I believe they use the same Sony sensors as Nikon?), the Fuji images were better. Even with the same depth of field, the Fuji images seemed like they had more dimension to them, and the tonal range in the highlights and shadows were better.
I have no idea how the Pentax would compare to the D810 because I know exactly zero about the D810, so maybe those images you posted would also look better next to the K5, but they still look different from the Fuji images for the same reasons.
(He sold the Pentax, btw. And whenever I decide to get a "real" digital camera instead of the dinky p&s I have for messing around, I plan on buying Fuji.)