In the book by Demazure "Lectures on $p$-Divisible Groups" a formal group functor over a field $k$ is defined in II.4 as a functor $\operatorname{Mf}_k \to \operatorname{Grp}$ where $\operatorname{Mf}_k$ is the category of finite $k$-rings.
In the (English translation of the) book by Zink "Cartier Theory of Commutative Formal Groups" a formal group over a commutative ring $K$ is defined in II.2.2 as an exact functor $\operatorname{Nil}_K \to \operatorname{Grp}$ commuting with infinite direct sums, where $\operatorname{Nil}_K$ is the category of nilpotent commutative $K$-algebras.
How are these definitions related when $K = k$? A $k$-ring cannot be a nilpotent $k$-algebra as it contains $1$.
