Pristine single-walled carbon nanotubes (SWCNTs) are rather inert to O$_2$ and N$_2$, which for low doses chemisorb only on defect sites or vacancies of the SWCNTs at the ppm level. However, very low doping has a major effect on the electronic properties and conductivity of the SWCNTs. Already at low O$_2$ doses (80 L), the X-ray photoelectron spectroscopy (XPS) O 1s signal becomes saturated, indicating nearly all the SWCNT's vacancies have been oxidized. As a result, probing vacancy oxidation on SWCNTs via XPS yields spectra with rather low signal-to-noise ratios, even for metallicity-sorted SWCNTs. We show that, even under these conditions, the first principles density functional theory calculated Kohn-Sham O 1s binding energies may be used to assign the XPS O 1s spectra for oxidized vacancies on SWCNTs into its individual components. This allows one to determine the specific functional groups or bonding environments measured. We find the XPS O 1s signal is mostly due to three O-containing functional groups on SWCNT vacancies: epoxy (C$_2$$>$O), carbonyl (C$_2$$>$C$=$O), and ketene (C$=$C$=$O), as ordered by abundance. Upon oxidation of nearly all the SWCNT's vacancies, the central peak's intensity for the metallic SWCNT sample is 60\% greater than for the semiconducting SWCNT sample. This suggests a greater abundance of O-containing defect structures on the metallic SWCNT sample. For both metallic and semiconducting SWCNTs, we find O$_2$ does not contribute to the measured XPS O~1s spectra.