In order to investigate the formation of O I 7771-5 and [O I] 6300/6363 lines, extensive non-LTE calculations for neutral atomic oxygen were carried out for wide ranges of model atmosphere parameters, which are applicable to early-K through late-F halo/disk stars of various evolutionary stages. The formation of the triplet O I lines was found to be well described by the classical two-level-atom scattering model, and the non-LTE correction is practically determined by the parameters of the line-transition itself without any significant relevance to the details of the oxygen atomic model. This simplifies the problem in the sense that the non-LTE abundance correction is essentially determined only by the line-strength (W-lambda), if the atmospheric parameters of T-eff, log g, and xi are given, without any explicit dependence of the metallicity; thus allowing a useful analytical formula with tabulated numerical coeffcients. On the other hand, our calculations lead to the robust conclusion that LTE is totally valid for the forbidden [O I] lines. An extensive reanalysis of published equivalent-width data of O I 7771-5 and [O.] 6300/6363 taken from various literature resulted in the conclusion that, while a reasonable consistency of O. and [ O.] abundances was observed for disk stars (-1 less than or similar to [Fe/H] less than or similar to 0), the existence of a systematic abundance discrepancy was confirmed between O I and [O I] lines in conspicuously metal-poor halo stars (-3 less than or similar to [Fe/H] less than or similar to -1) without being removed by our non- LTE corrections, i.e., the former being larger by similar to0.3 dex at -3 less than or similar to [Fe/H] less than or similar to -2. An inspection of the parameter-dependence of this discordance indicates that the extent of the discrepancy tends to be comparatively lessened for higher T-eff/log g stars, suggesting the preference of dwarf (or subgiant) stars for studying the oxygen abundances of metal-poor stars.