Let $X$ be smooth complex projective variety. Let $\mathcal{O}_{X}^{*}$ be the sheaf of non-vanishing morphisms to $\mathbb{C}$. Then we have the famous phrase "$H^{1}(X,\mathcal{O}_{X}^{*})$ is the group of isomorphism classes of line bundles on $X$".
I have a question about the precise meaning of this. Suppose we have $2$ line bundles $L_{1},L_{2}$ (in the algebraic category), whose set of transition maps are Cech $1$-cocycles, $\alpha_{1},\alpha_{2}$, which have the same class in $[\alpha_{1}] = [\alpha_{2}] \in H^{1}(X,\mathcal{O}_{X}^{*})$. Then can we find an isomorphism of line bundles which lifts the identity map. Namely a morphism $\phi: Tot(L_{1}) \rightarrow Tot(L_{2})$, which restricts to a linear isomorphism on each fibre of the total space?
note that there is a more general notion of isomorphism of line bundles, namely a morphism $\phi: Tot(L_{1}) \rightarrow Tot(L_{2})$, which maps each fibre to another fibre by a linear isomorphism.