In different articles I have seen different definitions of Barsotti-Tate representations. I am wondering if and how these definitions are equivalent.
In Section 1.1 of Conrad-Diamond-Taylor they say that, for $K/\mathbb{Q}_\ell$ a finite extension and $E\subset \overline{\mathbb{Q}}_\ell$ some coefficient field and $V$ a 2-dimensional vector space over $E$, a Galois representation $$ \rho: G_K \to GL(V) $$ is Barsotti-Tate if there exists an $\ell$-divisible group $\Gamma_{/\mathcal{O}_K}$ such that $\rho$ acting on a stable $\mathcal{O}_E$-lattice inside $V$ is isomorphic as a $\mathbb{Z}_\ell[G_K]$-module to the generic fibre of $\Gamma_{/\mathcal{O}_K}$ with its action by $G_K.$ (This definition explains the name.)
In Section 2.3.1 of Gee-Liu-Savitt they say that the representation $\rho$ as above is Barsotti-Tate if for all embeddings $\tau:K\to \overline{K}$ we have that the $\tau$-labelled Hodge-Tate weights are $HT_\tau(V)=\{0,1\}.$
It is not obvious to me that these definitions are equivalent. Is this easy to see? Is a reference for this fact available?
Thanks.