There's always an argument about Nyssa sylvatica/biflora trees and wetland/upland presence. Best to be cautious and open to exceptions, imo.
My personal impression of Nyssa and persimmon leaf arrangements is that the former looks spiraled like a green dragon (Arisaema dracontium) and the latter looks like soldiers in lines on either side of the stem (e.g. crepe myrtle). The Nyssa leaves vary greatly in texture, color, and shape while the persimmon leaves I see are all the same - kind of shaped like a papaya.
My guess is that persimmon bark has vertical striations broken by more horizontal fractures (like Southern live oak bark has so many of, and which are consequently sometimes compared to the blockiness of alligator skin) than the Nyssa, which also seem to have subtler striations. An older specimen would likely have more relief in the bark, of course.
It's hard for me to tell based on these pictures, but I suspect the bark is the key. Also, when it's hard to tell, it's usually a Nyssa... Not saying that that one is, but it's worth considering seriously.
What are the consequences of being mistaken in this case?