Keep in mind that Oxidation Is Loss of electrons, and Reduction Is Gain of electrons (OIL-RIG, LOL), A compound that is oxidized is by definition a reducing agent that reduces the other reactant (namely the oxidizer which is itself reduced in the process), and visa versa.
Oxygen is not needed (Riggs is wrong in his statements), only an electron willing to move from compound A to B is needed.
Then it gets more complicated from there. An oxidizer to one compound, can be a reducing agent to another compound.