From Lemma 16 of the paper "Vector Bundles over an Elliptic Curve" we can restrict to the following case: given a line bundle $\mathcal{O}_{E}(x)$ on an elliptic curve $E$ of degree 1 or 0, there exists a unique rank two vector bundle $V$ (up to isomorphism) given by the extension
$$0 \to \mathcal{O}_{E} \to V \to \mathcal{O}_{E}(x) \to 0.$$
Here, if the degree is zero, we must further assume the line bundle has a non-trivial section. In addition, by $x$ I just mean whatever point or points we are twisting by.
My question is, how do we actually compute $V$ in practice? Do we know under what conditions it splits as a direct sum $V \cong \mathcal{O}_{E} \oplus \mathcal{O}_{E}(x)$? Perhaps someone can show me a simple example where $V$ is constructed explicitly.