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Basic Answer:

The terms "electron-phonon interaction" and "phonon-electron interaction" mean the same thing and one almost always hears the former and not the latter. In a nutshell the term refers to the fact that the usual idea of separating the quantum system of electrons and the quantum system of vibrations (phonons) is an approximation that does not answer questions about the exchange of energy between the two systems. The next most sophisticated treatment involves including a term in the Hamiltonian that approximates the mechanism for that energy exchange. That term is called the electron-phonon interaction term.

Addendum on Electron-Phonon Interaction:

If the question were posed asking to explain the electron-phonon interaction, this answer would have to discuss the process of calculating electronic energies for fixed nuclei and then solving the problem where the nuclei are allowed to move within the adiabatic approximation. That leads to the two quantum systems mentioned above and thus to the need for an improvement which treats electrons and nuclei both being treated at the same time in a quantum mechanical fashion. One proceeds with an electronic Hamiltonian and a nuclear coordinate Hamiltonian and adds a third term called the electron-phonon interaction which is meant to be a good approximation to the full quantum system and is amenable to reasonable approximation methods.

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