= Solution
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.
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