Nothing-at-Stake is a theoretical problem invented by Vitalik so that he can promote his Casper PoS mechanism. It’s a far-fetched problem without much connection to reality. It has never been observed in practice in NXT and Ardor, we haven’t even seen the software which is supposed to be used by the forgers (the block generators, miners) to build on more than one chain. Chances are such software is not even developed, let alone adopted by a significant part of the stake holders. Why forgers don’t forge on more than one chain? The inventor of the theory should answer that, but probably because there is no incentive to do so, considering the possible losses. The forgers are afraid to not lose the value they have (which is fundamental part of the PoS economics), and without block reward, the profits from forging are very small. In fact so small, that many stake holders don’t forge at all.
BTW, IIRC, Ethereum was supposed to switch to Casper in 2017, then last autumn they said 2018. It is mid 2018 now. Seems they cannot solve the problem they invented themselves.
Generally, congratulations for understanding what nothing-at-stake is. Many authors wrongly use the term “nothing-at-stake” to describe the long-range attack, or other things.
NXT clients simply look for the chain with the most transactions.
No, this part is not true. The number of transactions is not used for fork resolution. Where did you read that?
If NXT does anything to solve N@S, it is to not have a block reward.