Hope I didn't skim too fast and miss too much of this discussion...
The guts of ENERGY is that it is not a "thing", it is an abstraction. That's partly what folks mean when they say, "it's a useful concept for analysing physical systems that is conserved, etc., etc.". It's also what Feynman might have meant when stating that we cannot say what energy is. That is not a mysterious utterance, it is entirely sensible once we realize that energy is an abstract concept.
Going back to the thread introduced by Mueinz and elaborated by TobyC and SpectraCat, the conservation of total energy is an expression of time translation symmetry for Lagrangian systems. And don't worry about systems that do not have a Lagrangian, since everything classically can be reduced to Lagrangian mechanics. You will want to worry about the implications of general relativity and quantum mechanics, but that is almost a whole other topic which has been touched upon but I won't elaborate on it here, other than to remark that in quantum mechanics not all that much changes, a systems total energy is represented by a Hamiltonian operator which generates time translation, which squares with the classical concept since if the Hamiltonian is constant there will be time symmetry.
If you want to get really philosophical, then everything talked about in physics is an abstraction. We don't really know what mass is, or what charge is, or what space is. All such terms are words we use to describe the world in a semi-objective communicable fashion. However, compared to concepts like mass and charge, energy is on a whole other level of abstraction. This is evident in the way energy can be transformed, KE <-> PE. If a thing can be transformed like this then you know you are not talking about something concrete.
To echo other contributors, this does not mean that energy is ill-defined. On the contrary, the fact that we can track energy, and show that for time symmetric systems the total energy is conserved, means we have a good grasp and definition of the concept.
An appropriate analogy I think is that trying to define what energy is, is like trying to define what money is. You cannot define money as a coin or paper note, since that doesn't cover other forms of payment for goods etc., money is indeed just about as elusive a concept as energy, yet we all know what we mean by money and we use it and try to conserve it everyday without any philosophical qualms about what it really is. It isn't anyone thing, it's a pure abstraction, yet any manifest form of which can be a concrete reality, such as a minted coin, just like any particular form of energy can be concrete, like KE being simply mass and velocity. Caveat: energy is just that bit more subtle than money, since energy is really only a useful concept when referring to changes in energy.
And there, finally, is another clue that you're dealing with a pure abstraction. Changes in energy are the important thing, the particular value of energy itself is meaningless. So once again, energy cannot be any special sort of "thing" since it's value is meaningless. But since changes in energy are extremely meaningful and indeed just about supply a complete way to describe physical motions, you have again the idea that energy is an abstraction, it is a relationship between things, but it is not a thing itself.
That makes me think of other analogies. Is "being taller than a giraffe", or, "being to the left of someone", a thing that you can define and grasp? Yes, of course. But these are not something you can grab a hold of and put in a bottle. They are not things in themselves, but are rather relationships between things. (I think the technical term in philosophy for a concrete "thing" is an "entity", but I don't see much point in worrying about such wording trifles). Such is the way with energy. We need to disabuse ourselves of the notion that when we heat a glass of water we are adding energy like some sort of substance. We are not. We are raising the average kinetic energy of the molecules .
So "Adding heat" = "Raising average KE".
The RHS of this equality of meanings is the correct way to think, the LHS is not quite bogus, but it can be misleading if you interpret it too literally.
So in reality we are not actually adding anything to the glass of water. It is only on some abstract account sheet that we can say we've raised the heat content or "added energy". And that's exactly the point, it's just an accounting ledger system. That's all energy is, however incredibly useful as a unifying concept in physics. Energy is not representing an actual thing we add to the water to make it hotter, it represents instead a change in state of the water that we induce, so it is a relationship between states of a system.
