That's the thing with physics, often you need to sacrifice precise definition so that you can explain something intuitively to someone who is new to the subject.
For example, we are all taught that the electron orbits round the nucleus. But when we learn QM, we find that actually its much more complicated, since the electron doesn't take a particular path through space. And the s-orbital electrons have zero orbital angular momentum, so they're not even moving around the nucleus at all!
But back to energy, I've found (over my time learning physics) that the precise meaning of energy depends on the branch of physics you are using it in.
For lagrangian mechanics, energy is a conserved quantity of the system as long as the potential of the system is not explicitly time-dependent.
In QM, the energy eigenstates are stationary states, meaning that if the entire system is an energy eigenstate, then a change in time only changes the state by a phase factor. (Which is due to the time dependent Schrödinger equation).
Energy/mass in Einstein's relativity is a generalisation of classical energy, such that it is a locally conserved quantity.
Energy in statistics is a useful concept, in that it gives us the distribution of the number of particles in particular energy states.
Energy density in electrodynamics, denoted [itex]u[/itex] is given by the equation:
[tex]\frac{\partial u}{\partial t}=- \nabla \cdot S - J_f \cdot E[/tex]
Application of Noether's theorem says that energy conservation is due to invariance with respect to time translation. So the energy conservation is due to this time symmetry, which is mathematically represented by some kind of group.
This last definition seems like the best and most general definition of energy. Unfortunately, I don't know much about the maths of groups, etc, so I can't tell you exactly what the group is or how it gives rise to a definition of energy conservation..
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