Reading from Introduction to Nonlinear Laser Spectroscopy - Revised Edition (1988) by Marc D. Levenson and Satoru S. Kano (p13-14, bold emphasis mine):
Consider a medium in which each atom can absorb only one definite frequency, but in which different atoms can absorb different frequencies. Such a situation can result, for example, from the Doppler effect which allows atoms moving away from a light source to absorb a frequency below that absorbed by atoms moving toward the source. The absorbed frequencies, however, are the same in the rest frame of each atom. The absorption lines of such a medium are termed inhomogeneously broadened, because different atoms are responsible for different portions of the absorption line. If any frequency within the absorption band could excite every atom with the same probability, the line would be homogeneously broadened. Such a situation can occur in an atomic beam where the atomic velocities are perpendicular to the direction of propagation of the light .
I don't understand what the bold text is referring to. It seems there must be a rest frame in which the absorbed frequencies are different, or there would not be inhomogeneous broadening (from the perspective of the instrument). In what sense are the absorbed frequencies the same?