Let $K$ be a nonarchimedean local field with residue field $k$ of characteristic $q = p^N$, and pick a uniformizer $\pi\in \mathscr{O}_K$. Recall that explicit local class field theory, à la Lubin--Tate, constructs a totally ramified extension $K_\pi$ of $K$ as follows. Let $f\in \mathscr{O}_K[[x]]$ be a formal power series such that $f(x)\equiv \pi x \pmod {x^2}$ and $f(x)\equiv x^q\pmod \pi$. Then one can construct a formal group law $F_f(x,y)$ over $\mathscr{O}_K$, which admits the structure of a formal $\mathscr{O}_K$-module with $[\pi](x) = f(x)$.
The maximal ideal $\mathfrak{m}$ of the ring of integers of $K^\mathrm{sep}$ has the structure of an abelian group, with group operation given by $F_f(x,y)$. Let $\mathfrak{m}_{\pi^\infty}$ denote the $\pi$-power torsion submodule of $\mathfrak{m}$. One then defines $K_\pi = K[\mathfrak{m}_{\pi^\infty}]$, and proves that $K_\pi\cdot K^\mathrm{ur} = K^\mathrm{ab}$.
Along the way, one shows that different choices of $f(x)$ give rise to isomorphic formal group laws $F_f(x,y)$. Picking a formal power series $f(x)$ is therefore like picking a coordinate on some formal group scheme $G_\pi$ over $\mathscr{O}_K$. This motivates my question: is there a direct way of constructing this formal group scheme $G_\pi$ solely from the data of $K$ and $\pi$ (without having to pick $f(x)$)? How do different choices of $\pi$ affect $G_\pi$? What conditions on a formal group over $\mathscr{O}_K$ guarantee that it arises in this manner?
It seems like the answer should be yes, but I could not find such a construction. If anyone has a reference, that would be much appreciated!
Some thoughts: after picking $f(x)$, the formal group law $F_f(x,y)$ is a deformation of the formal group law $F_f(x,y)\pmod\pi$ of height $M = N\cdot v_\pi(p)$ over $k = \mathscr{O}_K/\pi$. In particular, since the Lubin-Tate ring $R_\mathrm{LT}$ classifies deformations of formal groups, I get a map $R_\mathrm{LT} \to \mathscr{O}_K$ of local rings. Writing $R_\mathrm{LT}$ (non-canonically!) as $W(k)[[u_1,\cdots,u_{M-1}]]$, one can explicitly write down (in terms of the coefficients of $f(x)$) what each $u_i$ is sent to. Perhaps one can show that different choices of the $u_i$s correspond exactly to different choices of $f(x)$ (e.g. by using the images of the $u_i$s to construct the formal group law, and then using the logarithm to get some endomorphism $f(x)$ satisfying the appropriate conditions)? (Even if this approach worked, this wouldn't be very satisfying.)
