Dead nut on. Fluorescence is simply the release of energy from an atom in response to energy added to the atom. The amount of energy in equals the amount of energy out. The energy out, however, is a longer wavelength (lower energy) photon. The rest of the energy is released as radiative or non-radiative energy. In non-radiative release the energy is transferred to something else, like the walls of the container, as in a fluorescent light or it can be radiative by releasing a photon at each transition back to ground state.
In a YAG laser, the YAG rod actually fluoresces. It absorbs photon radiation at several wavelengths via a lamp or at 808nm if diode pumped and releases that energy at 1064nm as well as some other wavelengths. A few photons of course happen to be released parallel to the cavity and gain begins resulting in a 1064nm laser beam but there is quite a bit of fluorescent light generated that escapes the rod.