Changing the two local frames by the nowhere-zero functions $\psi_j$ replaces the transition function $f$ by
$$
\frac{\psi_1}{\psi_0}f=z^k.
$$
With the standard convention that the <tautological bundle> $\mathcal O_{\mathbb{CP}^1}(-1)$ has transition function $z$, its $k$th tensor power $\mathcal O_{\mathbb{CP}^1}(-k)$ has transition function $z^k$. Thus
$$
L\cong\mathcal O_{\mathbb{CP}^1}(-k).
$$
This is the <smooth classification of complex line bundles on the complex projective line> by their winding number.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 115 / 2 / i / Solution

{title2=$L\cong\mathcal O_{\mathbb{CP}^1}(k)$}

Every smooth complex line bundle on $\mathbb{CP}^1$ is isomorphic to exactly one twisting bundle $\mathcal O(k)$. Trivializing on the two standard affine charts leaves a transition function on $\mathbb C^*$; its <winding number> determines $k$, and a <partition of unity> removes the zero-winding factor.

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 Incoming links: Smooth classification of complex line bundles on the complex projective line