The idea that excitonic state (electronic) coherences are of fundamental importance to natural photosynthesis gained popularity when, a decade ago, slowly dephasing quantum beats were observed in the two-dimensional electronic spectra of the Fenna-Matthews-Olson complex at 77 K. These were assigned to quantum superpositions of excitonic states; a controversial interpretation, as the spectral linewidths suggested fast dephasing arising from strong interactions with the environment. While it has been pointed out that vibrational motion produces similar spectral signatures, concrete assignment of these coherences to distinct physical processes is still lacking. Here we revisit the coherence dynamics of the Fenna-Matthews-Olson complex using polarization-controlled two-dimensional electronic spectroscopy, supported by theoretical modelling. We show that the long-lived quantum beats originate exclusively from vibrational coherences, whereas electronic coherences dephase entirely within 240 fs even at 77 K - a timescale too short to play a significant role in light harvesting. Additionally, we demonstrate that specific vibrational coherences are excited via vibronically coupled states. The detection of vibronic coupling indicates the relevance of this phenomenon for photosynthetic energy transfer.