The 14 m telescope at FCRAO was used to survey the CO J = 1-0 line emission from all M giants, S stars, and C stars in the Two Micron Sky Survey having strong far-infrared emission [Fν(60 μm) > 15 Jy] and lying north of δ = -10°. Combining the data obtained here with observations of other carbon stars obtained both at FCRAO and with other telescopes shows that the line profiles from carbon and S-type stars with optically thick outflows look like flattened parabolae and have roughly the same shape as each other, independent of the telescope with which they were observed. The data also suggest that many of the carbon stars in our sample are resolved by the 45″ beam of the FCRAO telescope. A quantitative model is developed which accounts for these phenomena. In contrast, the shapes of the spectral lines from giant M stars are diverse, ranging from triangular to spiked and asymmetric. Only a few line profiles look like flattened parabolae. These results suggest that the envelopes of M stars have complex kinematics and structure. The line widths of M stars, and the ratio of CO brightness to far-infrared continuum brightness for M stars, are each less than those for carbon stars. The interpretation of these differences is complicated by the complex structures of the M star envelopes. These results suggest that mass-loss rates of oxygen-rich stars may not be accurately deduced from their far-infrared fluxes using the algorithm developed from studies of carbon stars. Furthermore, the differences in the observed CO line shapes and corresponding envelope structures of M and C stars suggest that, if most M giants evolve into C stars, the change must be accompanied by a change in the mechanism driving the mass loss. The outflow velocities inferred from the line profiles of the stars span a range of more than an order of magnitude. C star velocities are correlated with infrared color. Since our sample is, in effect, color-selected, the average outflow velocity for all C stars within 1.5 kpc of the Sun is probably significantly less than the mean velocity of our sample of C stars, 17 km s-1.