Pendulum with a rotating point of support from Landau-Lifschitz

Let's compare Landau's Lagrangian and the one given by $\mathcal L_{Me}$ in your question:

$$\mathcal{L}_{Landau}-\mathcal{L}_{Me}=mal\gamma^2\sin(\phi-\gamma t)-alm\gamma\dot\phi\sin(\phi-\gamma t)=\\ =mal\gamma\left((\gamma-\dot\phi)\sin(\phi-\gamma t)\right)=\\ =mal\gamma \frac{\text{d}}{\text{d}t}\cos(\phi-\gamma t)$$

Now the difference is obviously a total time derivative, thus you haven't made a mistake, just undersimplified the Lagrangian.