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Based on micromagnetic simulations and experimental observations of the magnetization and lattice dynamics following the direct optical excitation of the magnetic insulator Bi:YIG or indirect excitation via an optically opaque Pt/Cu double layer, we disentangle the dynamical effects of magnetic anisotropy and magnetoelastic coupling. The strain and temperature of the lattice are quantified via modeling ultrafast x-ray diffraction data. Measurements of the time-resolved magneto-optical Kerr effect agree well with the magnetization dynamics simulated according to the excitation via two mechanisms: The magneto-acoustic coupling to the experimentally verified strain dynamics and the ultrafast temperature-induced transient change in the magnetic anisotropy. The numerical modeling proves that for direct excitation both mechanisms drive the fundamental mode with opposite phase. The relative ratio of standing spin-wave amplitudes of higher order modes indicates that both mechanisms are substantially active.