I've been trying to see how both definitions of a formal group coincide:
Take a ring $ R $, a group object in the category of formal schemes over $ R $ is then given by two morphims $ e : R \rightarrow X $, $ \mu : X \times X \rightarrow X $.
For $ X = \widehat{\mathbb{A}}^1_R $, this amounts to two morphisms of $R$-adic algebras $ e : R[[x]] \rightarrow R $ and $ \mu : R[[x]] \rightarrow R[[x,y]] $, but it is easy to see that $ \mathrm{Hom}_R(R[[x]], R) \simeq \mathrm{Nil}(R) $, so there might be more than $1$ possible choice for $ e $. Yet, in the definition of a formal group law, unitality is then stated as $ \mu(x) = x + y + \text{higher terms} $, but if for example $ R = \mathbb{Z}/4\mathbb{Z} $, take $ e : X \mapsto 2 $, then unitality reduces to $ \mu(x)(2, Y) = Y, \mu(x)(X, 2) = X $, but then $ \mu(x) = x + y -2 $ satisfies unitality and associativity.
In the case of $ R $ a field, there is of course no longer any problem, but I've seen the formal group law definition stated for $ R $ in this way.
So is it that we require formal group laws to actually have a fixed unit morphism?
I think in the context of formal group laws, one defines adic algebras over $R$ to be augmented $R$-algebra $\eta:A \to R.$ The topology on $A$ is the adic topology generated by $\text{Ker}(\eta),$ which is sometimes also assumed to be complete and separated. The maps between such adic algebras are required to be compatible with this data of the augmentation which fixes the problem you mentioned.