In this context atom means isolated atom, which has defined quantum states. This is to be distinguished from atom in a molecule, which is an entirely conceptual construction that has no strict definition or physical meaning. Atoms in a molecule have no defined quantum states.
That is, only an isolated oxygen atom outside 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 a 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.