Let $L$ be a $p$-adic number field, $\mathcal{O}_L$ its ring of integers, $\pi$ a uniformizer of $L$ and $q$ the cardinality of its residue field.
Let $\varphi(t)\in \mathcal{O}_L[[t]]$ be a Frobenius power series for $\pi$, meaning that $$ \varphi(t)=\pi t+t^2(\ldots) $$ such that $\varphi(t) \ \text{mod } \pi=t^q$. I am familiar with the definition of a Lubin-Tate formal group law $F_{\varphi}(X,Y)\in \mathcal{O}_{L}[[X,Y]]$ for $\varphi$ from these notes.
The most well-known example is when $L=\mathbb{Q}_{p}, \pi=p$ and $\varphi=(1+t)^p-1$, then $F_{\varphi}$ becomes $\mathbb{G}_{m}=X+Y+XY$.
I see though that many papers talk instead about Lubin-Tate groups instead of formal group laws and seem to call the above the above example as the multiplicative group. My question is what is the definition of a Lubin-Tate group then?
My guess is the following:
Using $F_{\varphi}$ one can define a group structure $(\mathcal{O}_{L},+_{F_{\varphi}})$, where $$ x+_{F_{\varphi}}y:=F_{\varphi}(x,y) $$ which can be checked to be a group from the axioms of a formal group law. Is this what the papers mean?
