Isomorphism of line bundles

It is useful to say that two line bundles $L \to M, J \to M$ are isomorphic if there is a diffeomorphism map $\varphi : L \to J$ such that $\varphi(L_m) \subset J_m$ for every $m \in M$ and such that the induced map $\varphi\vert _{L_m} : L_{m} \to J_{m}$ is a linear isomorphism.

We define a line bundle $L$ to be trivial if it is isomorphic to $M \times \mathbb{C}$ the trivial bundle. Any such isomorphism we call a trivialisation of $L$.