The meaning of transferring energy and storing energy I am taking a free online course on introductory physics, and I am having difficulty understanding the concept of energy. I believe a lot of the difficulty comes from the choice of words used, when talking about energy, so I hope you can confirm or deny or elaborate on my points below. I am a mathematician by training, but a layman in physics, so my questions and thoughts below are from that standpoint.
First off, if I raise an object some distance above ground-level, then I do work on the object, and one of the phrases that is being used is that I transfer energy to that object. My question is if is there an actual flow of energy coming from me to the object? Am I giving my energy away to the object?
I think the answer is no, and that the phrase “transfer of energy” just means that the energy that I spend or lose is equal to the energy that the object gains because of our interaction. (It may be unclear what exactly I mean by “flow of energy”, but I imagine something similar to water flowing from a place with a high water level to a place with a lower water level).
Second, when lifting an object and talking about gravitational potential energy, the phrases the energy is stored in the object or energy is held by the object is often used. Some webpages qualify that the energy is actually stored in the gravitational field, and not in the object itself. Similarly, when a spring is compressed, the spring is said to store potential energy, and some webpages say that the energy is stored in the bonds between its atoms.
My understanding is that a field is just the presence of a force, and as such, it should not have the ability to “store” anything? And in the same vein, do atomic bonds have the ability to store energy?
The thoughts that I have had so far, which make more sense to me, are this: The way we talk about energy is a convenience, which does not necessarily match how the energy concept really works. Energy is neither stored nor transferred. Rather, energy is a derived quantity, which is interesting because it is preserved within a closed system. From the state of a given object (i.e. given its speed, or its position in a field, or tensions within itself etc.), we may calculate a quantity that we call the object's energy. If the state of the object is changed, this may result in a change of the energy for the object, and if so, this increment or decrement in energy will match a decrement or increment of energy in the rest of the system, which can be useful in determining the new state of the system.
 A: The idea of energy 'flowing', as if it is an object, is as you said, not particularly useful.
Your paragraph starting "I think the answer is..."  seems right.
The last paragraph seems right too.
A few more comments:
It often seems as though the energy concept is just a useful calculating mechanism, that enables various properties of a system to be found.  Historically there was a theory that tried to treat energy and transfer of heat as the flow of a substance, see https://en.wikipedia.org/wiki/Caloric_theory
However another thing that could be considered is the equivalence of mass and energy $E=mc^2$, this could make us wonder again whether energy and fields storing energy should be treated as a real physical quantity that can 'flow'.
But for most everyday purposes physicists just use energy to help with calculations.
A: You are on the right track.
I like to think of energy as if it were money. Money in the bank represents stored work, which is energy that can be spent to perform work elsewhere as desired. Energy gets distributed between systems as work gets performed, and the form in which it winds up can change nature (it might start as chemical potential energy and wind up as kinetic energy of motion) which is like changing dollars for yen or pounds.
A: Transfer of energy is a concept that makes sense when there's a clear causal relationship between what happens to one object and what happens to another.  The sum total of the energy must remains the same, so if one object loses X energy and another gains X energy, we say that energy was "transferred."
There often is a "flow," but it happens astonishingly quickly.  If you watch slow motion footage of objects getting struck, you can see that a pressure wave forms, transferring the energy.
As for how energy is stored, its tricky because the answer is a bit of both.  It's stored in how the objects interact with the fields.  An object (or objects) in a field have "stored" energy which is energy defined by their position within that field.  The gravitational potential energy stored in an object as it is lifted up is defined by the objects position within the Earth's gravitational field.  As you raise it up, you increase it's potential.
One thing that may help is a pattern that shows up in the equations for energy.  What we find is that we can separate energy into "kinetic energy," which is depended only on the velocity of the particles and not their positions ($KE(v)$), and "potential energy," which depends only on the position of the particles and not their velocity ($PE(x)$).  We can sum these together to get total energy, but what's interesting is that it is separable in this way.  We have $E=KE(v)+PE(x)$, but we can generally separate them and adding the results together.  You don't get stuck with some $E=KE(x, v) + PE(x, v)$ or anything like that.
Well, it does get more complicated than that, but you can get suspiciously far with just those two classes of energy, and addition!  This idea that potential energy is related to the positions but not the velocities is really useful for understanding things.
