Here, it states the four equivalent conditions for a system of parameter without being Noetherian. I can show the equivalence between 1, 2 and 4 without using Noetherian property. But I can not deduce 3. When it is Noetherian, modulo out the ideal and use Noetherian ring's nilradical is nilpotent. But without the Noetherian property, it might not work?
Is it wrong?
It is indeed wrong for non-Noetherian rings, and you can easily find a counterexample using your observation that it depends on the nilradical being nilpotent. For instance, let $k$ be a field and let $R=k[x_1,x_2,x_3,\dots]/(x_1,x_2^2,x_3^3,\dots)$. Then $R$ is local with maximal ideal $(x_1,x_2,x_3,\dots)$ which is also the nilradical. So, the empty system satisfies 1, 2, and 4, but does not satisfy 3 since the maximal ideal is not nilpotent.