= Solution
Take the drift propagator with the correct initial normalization, $U_0(t,t_0)=e^{-i(t-t_0)H_0}$, and define the <interaction picture> by $U=U_0U_I$. Differentiate this product and substitute the <Schrödinger equation>:
$$
i\dot U_0U_I+iU_0\dot U_I=\left(H_0+\sum_m f_mH_m\right)U_0U_I.
$$
Since $i\dot U_0=H_0U_0$, the drift terms cancel. Left multiplication by $U_0^\dagger$ gives
$$
\boxed{i\dot U_I=\sum_m f_m(t)\widetilde H_m(t)U_I,\qquad \widetilde H_m=U_0^\dagger H_mU_0,\qquad U_I(t_0,t_0)=I.}
$$
The printed $e^{-itH_0}$ is this expression with time origin $t_0=0$. Its differential identity remains valid at arbitrary $t_0$, but its interaction-picture initial operator would then be $e^{it_0H_0}$ rather than the identity. Using $t-t_0$ resolves that normalization without changing the derivation.
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