The conversation does tend to wander about rather aimlessly. So let's tackle some of the basics.
In relation to defining the energies....
Energy - The capacity of a system to do work.
By this, if it has thermal energy, it is capable to do work in relation to exertion of heat. Likewise kinetic energy, it is capable to do work due to its motion.
etc....
Kinetic energy- the energy an object has due to its motion. Dependant on mass and velocity.
Potential energy- the energy stored within a system that can be released at a later point.
Work - the exertion of energy.
any of these are up for debate
The definition of energy as "the capacity of a system to do work" is a standard textbook definition, at least in high school text books and maybe some basic college texts, but it isn't useful for much of anything other that answering the question on a test, "What is energy?"
As for the definition of work as "the exertion of energy" that's a new one to me. Perhaps it's from a text book, but wherever it comes from it's rather worthless. When combined with the previous definition of energy, it all sounds circular. Work is properly defined as the line integral of a force along a given path. If the force is constant and the direction of the force and the direction of motion are parallel, then the line integral reduces to a common high school level definition of force times distance. Consider some simple cases:
1.The work to lift a mass m to a height h - Gravity is exerting a downward force of mg, so to lift it requires an upward force of mg. Therefore since the force is constant the work is mg times h = mgh.
2. The work to move a mass m a distance of l on a horizontal surface with coefficient of friction u. The normal force is the weight, mg. The force of friction that must be overcome is therefore mgu and the work is mgul.
An important difference between 1 and 2 is that the work done in case 1 is can be recovered by lowering the mass back to its starting point. In case 2 it takes more work to return the mass to its starting point.
The force in case 1 is said to be conservative while the force in case 2 is said to be non-conservative.
Potential energy can only be defined for conservative forces. Potential energy is just the work requires to move a system from a reference point to its final point. The reference point is arbitrary. Only differences in potential are meaningful. For gravitational potential near the earth where the earth may be treated as flat the reference point is the surface of the earth. Thus the potential energy of a mass m at a height h is just mgh, the work required to lift it from the surface. When the curvature of the earth is important, as in the case of an orbiting satellite the reference is chosen to be infinity.
Kinetic energy is the energy due to motion. There is both linear and rotational kinetic energy. Linear kinetic energy is given by (mv^2)/2, while rotational kinetic energy is given by (Iω^2)/2 where I is the moment of inertia and ω is the angular velocity.
The energy of a system is simply the sum of its various types of energy with each type of energy calculated in a specific way. The importance of the energy concept and the reason that it is used is that it obeys a conservation law. You calculate the energy of a system at the beginning of a process, then the system goes through some process and at the end when you compute its energy the number is the same as it was at the start. The energy serves to determine what processes can take place out of all imaginable processes. Now some may find this definition of energy unsatisfying; however this is the way that energy is understood in physics and the only reason that it is important.
If only mechanical problems are considered then the energy concept is not needed. Using energy methods can shorten a calculation but they are not necessary. All problems can be solved using forces alone. With the rise of the industrial age when people started burning fuel to make steam to run a steam engine that produced mechanical work, the problem became how to obtain the most work for the least fuel. It was this problem that lead to thermodynamics and the formulation of the law of conservation of energy. The basic equation is one that connects quanties of work, internal energy and heat. At about the same time electromagetism was understood, and the concept of energy in the electromagnetic field was developed. With special relativity yet another term was added to the energy equation.