Here's an argument that Eric Peterson and I came up with showing that the homotopy type of Morava $E$-theory only depends on the choice of perfect char $p$ field $k$ and the height $n<\infty$.
$\textbf{Lemma}$: Let $R$ be a ring with two Landweber exact formal group laws $e,f:\text{Laz}\rightarrow R$ and let $E$ and $F$ be the spectra corresponding to these two formal group laws under the Landweber exact functor theorem. Then $E$ and $F$ have the same homotopy type if there is a ring extension $u: R\rightarrow S$ which is split as an $R$-module map, and such that the formal group laws $u\circ e$ and $u\circ f$ are isomorphic and Landweber exact.
$\textbf{Proof}$: First we show that $E\simeq F$ if the map $\eta:E_*\rightarrow F_*E$ (induced by $1\wedge id: \mathbb{S}\wedge E\rightarrow F\wedge E$) is split (as a map of $F_*$-modules). Note that $E_*$ is canonically an $F_*$ module by the \textit{equality} $E_*=R=F_*$. We claim that $F_*E$ is flat over $F_*$. That boils down to interpreting Landweber exactness as flatness of maps to $\mathcal{M}_{fg}$, noting that flatness is preserved under pullback, and that (Spec of) $E_*$, $F_*$ $F_*E$ and $\mathcal{M}_{fg}$ fit into a pullback square (with $\mathcal{M}_{fg}$ in the bottom right and Spec of $F_*E$ in the top left). It follows that $F\wedge E$ represents the cohomology theory $X\mapsto F^*X\otimes_{F_*}F_*E$, and then Brown representability (and Strickland's result about the absence of phantom maps between Landweber exact spectra) produces a map $\phi: \text{Mod}_{F_*}(F_*E,F_*)\rightarrow F^*E$. Now suppose $s$ is a splitting of $\eta$. Then there is a map $\phi(s):E\rightarrow F$ which on homotopy groups (set $X=\text{pt}$ above) is the composite $s\circ\eta$ which is the identity map (recall that $E_*$ and $F_*$ are canonically isomorphic to $R$), so $E\simeq F$.
Now we show that $\eta:E_*\rightarrow F_*E$ is split if there is a ring extension $u: R\rightarrow S$ which is split as an $R$-module map, and such that the formal group laws $u\circ e$ and $u\circ f$ are isomorphic and Landweber exact. Write $g:=u\circ e$ and $h:=u\circ f$. Assume they are isomorphic and Landweber exact. Let $G$ and $H$ be the spectra associated to $g$ and $h$. Then we have the following commutative diagram of $F_*(=R)$-modules
$\require{AMScd}$
\begin{CD}
S=G_* @>1\otimes 1\otimes id>> H_*G= S\otimes_{MU_*}MU_*MU\otimes_{MU_*}S\\
@A u A A @AA u\otimes id\otimes u A\\
R=E_* @>>1\otimes 1\otimes id> F_*E=R\otimes_{MU_*}MU_*MU\otimes_{MU_*}R
\end{CD}
Since the right vertical map is split (as a map of $F_*$-modules), splitting the bottom horizontal map (as a map of $F_*$-modules) is equivalent to splitting the diagonal map (as a map of $F_*$-modules). Moreover, the top horizontal map is split (as an $S=H_*$-algebra (and hence $F_*$-algebra) map in fact) by first choosing an isomorphism of $g$ and $h$, which induces an $H_*$-algebra isomorphism $H_*G\simeq G_*G$ and then using the multiplication map $G\wedge G\rightarrow G$ to split $G_*\rightarrow G_*G$. Therefore splitting the diagonal map is equivalent to splitting the left vertical map (as a map of $F_*$-modules).
$\textbf{End of proof}.$
Now we apply the lemma to show that at height $n<\infty$ the homotopy type of Morava $E$-theory depends only on the choice of a perfect characteristic $p$ field. In other words, let $k$ be a perfect characteristic $p$ field and let $E_1(n)$ and $E_2(n)$ be two Morava $E$-theory spectra corresponding to two arbitrary height $n< \infty$ formal group laws over $k$. Then we'll show that as spectra, $E_1(n)\simeq E_2(n)$.
It suffices to check the conditions of the lemma. Let $\overline{k}$ be the algebraic closure of $k$. Let $R$ and $S$ be the (2-periodic) Lubin-Tate deformation rings $W(k)[[u_1,...,u_{n-1}]][\beta^\pm]$ and $W(\overline{k})[[u_1,...,u_{n-1}]][\beta^\pm]$. Let $\tilde{e}$ and $\tilde{f}$ be formal group laws over $R$ universally deforming $e_1$ and $e_2$. Then $\tilde{e}_1$ and $\tilde{e}_2$ are Landweber exact and the corresponding spectra are the Morava $E$-theory spectra $E_1(n)$ and $E_2(n)$. Let $u:R\rightarrow S$ be the map that extends coefficients by $W(k)\rightarrow W(\overline{k})$, sends $u_i$ to $u_i$, and $\beta$ to $\beta$. Then the formal group laws $\tilde{g}:=u\circ \tilde{e}_1$ and $\tilde{h}:=u\circ \tilde{e}_2$ over $S$ are isomorphic and Landweber exact, since they universally deform the isomorphic formal group laws $g:u\circ e_1$ and $h:u\circ e_2$ over $\overline{k}$. Finally, the map $u:R\rightarrow S$ is split as an $R$-module map, because the map $W(k)\rightarrow W(\overline{k})$ is split as a $W(k)$-module map (See (ill Sawin's comments at Splitting the Witt vectors of $\overline{\mathbb{F}_p}$))
