Kelsey,
By definition, spitting water into H2 and O2 is not fusion. By definition, however, it is fission. It is not nuclear fission. As you pointed out, nuclear energy involves splitting atoms, not molucules.
From my non-nuclear physicist point of view, the two important concepts in a nuclear reaction are :1) converting matter to energy according to Einstein's relationship, 2) run-away (positive feeeback) reactions. When Uranium is bombarded with neutrons, the resulting split matter has less mass than the original. The lost mass is given off as energry. That is why a fist-full of Uranium can generate a massive amount of energy.
The energy that was released was the binding energy that held the original atom together. How is all this relevant?
If you are a physicist, then I have a question for you: is the binding energy/mass that is release when splitting an atom an equivalent concept to the binding energy released when you split a molecule? The magnitudes are obviously not the same, otherwise we would be splitting water (instead of Uranium) for nuclear energy.