In this content atom means isolated atom, which have defined quantum states. This is to be distinguished with atom in a molecule, which is an entirely conceptual construction that have not strict definition or physical meaning. Atoms in a molecule have no defined quantum states.
That is, only an isolated oxygen atom not in a molecule can have designations like $O(^3P)$, etc. For molecules, term symbols belong to the entire molecule, not individual atoms.
In oxidation reactions atomic oxygen is rarely involved. It can only be formed when very large amount of excess energy is available, such as in combustion or ignition processes, or in a radiation rich environment such as outer space. There, the majority of atomic oxygen should be the most stable species $O(^3P)$. However, in these high energy scenarios, all the other low lying states of oxygen will also be present in smaller quantities.