At the atomic level that's exactly what should happen. You start off with lithium metal, and up with a lithium ion. That's one electron per atom.
The voltage generated depends on the relative concentrations of uncharged lithium atoms and lithium ions (and the concentrations of the charge sink atom in the battery).
The energy you extract is product of the charge you move around - which is one electron per lithium atom - and the output voltage at the instant you moved it - and that goes down as you discharge the battery, and will vary (a bit) with temperarture.
You are changing one pair of chemical compounds into a different pair chemical compound. The electrical power you get out is the energy difference between the two sets of molecules.
The energy content of all four compounds depends on their temperature and their concentration. The dependence on temperature isn't dramatic, since it depends on the entropies of the four compounds involved - which is what is brought out in the Gibbs free energy equation. The dependence on concentration is what's meant by charging a battery. It isn't linear.
Explaining the result in chemical terms does depend on the supplier being explicit about the chemicals present in a charged battery, and the uncharged battery.
LiC6 + CoO2 <=> C6 + LiCoO2
The lithium moves through the battery, and and the electrons move through the electrical circuit.
Getting insight into what is going is pretty much second year university chemical thermodynamics, which is hard to get to grips with. This kind of forum doesn't lend itself to delivering the weeks of lectures that I had to sit through (and the stack of problems that I had to work through (twice as it turned out because once I'd got a proper grip of the subject, I realised that my first pass through the problems hadn't actually solved what I should have been solving for).