Oxygen abundances of 67 dwarf stars in the metallicity range -1.6 < [Fe/H] < -0.4 are derived from a non-LTE analysis of the 777 nm O I triplet lines. These stars have precise atmospheric parameters measured by Nissen and Schuster, who find that they separate into three groups based on their kinematics and alpha-element (Mg, Si, Ca, Ti) abundances: thick disk, high-alpha halo, and low-alpha halo. We find the oxygen abundance trends of thick-disk and high-alpha halo stars very similar. The low-alpha stars show a larger star-to-star scatter in [O/Fe] at a given [Fe/H] and have systematically lower oxygen abundances compared to the other two groups. Thus, we find the behavior of oxygen abundances in these groups of stars similar to that of the a elements. We use previously published oxygen abundance data of disk and very metal-poor halo stars to present an overall view (-2.3 < [Fe/H] < +0.3) of oxygen abundance trends of stars in the solar neighborhood. Two field halo dwarf stars stand out in their O and Na abundances. Both G53-41 and G150-40 have very low oxygen and very high sodium abundances, which are key signatures of the abundance anomalies observed in globular cluster (GC) stars. Therefore, they are likely field halo stars born in GCs. If true, we estimate that at least 3% +/- 2% of the local field metal-poor star population was born in GCs.