Atmospheric parameters and oxygen abundances of 825 nearby FGK stars are derived using high-quality spectra and a non-local thermodynamic equilibrium analysis of the 777nm OI triplet lines. We assign a kinematic probability for the stars to be thin-disk (P_1_), thick-disk (P_2_), and halo (P_3_) members. We confirm previous findings of enhanced [O/Fe] in thick-disk (P_2_>0.5) relative to thin-disk (P_1_>0.5) stars with [Fe/H]<~-0.2, as well as a "knee" that connects the mean [O/Fe]-[Fe/H] trend of thick-disk stars with that of thin-disk members at [Fe/H]>~-0.2. Nevertheless, we find that the kinematic membership criterion fails at separating perfectly the stars in the [O/Fe]-[Fe/H] plane, even when a very restrictive kinematic separation is employed. Stars with "intermediate" kinematics (P_1_<0.7, P_2_<0.7) do not all populate the region of the [O/Fe]-[Fe/H] plane intermediate between the mean thin-disk and thick-disk trends, but their distribution is not necessarily bimodal. Halo stars (P_3_>0.5) show a large star-to-star scatter in [O/Fe]-[Fe/H], but most of it is due to stars with Galactocentric rotational velocity V<-200km/s; halo stars with V>-200km/s follow an [O/Fe]-[Fe/H] relation with almost no star-to-star scatter. Early mergers with satellite galaxies explain most of our observations, but the significant fraction of disk stars with "ambiguous" kinematics and abundances suggests that scattering by molecular clouds and radial migration have both played an important role in determining the kinematic and chemical properties of solar neighborhood stars.