imok:
Lets take a step back and take a deep breath! Clear your preconceived thoughts, right or wrong. This is not as complicated as it is made out to be otherwise there won't be many electrical engineering graduates or only geniuses would be electrical engineers.
Also draw a boundary to limit our discussion in order to understand the basics. So we will not go beyond Newtonian physics, still valid for our day-to-day life. (Imagine if we try to explain all knowledge in one class! No one will learn a thing.)
If you re-write your original statements as "Electrical current is established when electrical energy flows. 'Electrical' energy is same thing as electricity" Then they would be more accurate. Read this a few times.
Also when you say electricity moves back and forth, you are thinking electrons or electrical charge as electricity that is not accurate. Electricity is the ‘name’ given to electrical form of energy, same as ‘heat’ is for thermal energy.
You can think of movement of electrons under the influence of electrical pressure/field same as vibration of atoms under influence of thermal gradient. More the ‘vibration’, more elevated is the temperature; we say more ‘heat’ is there. You don’t call those molecules ‘heat’. Heat ‘transfers’ or moves; molecules don’t.
As for AC and DC it gets little complicated but I will try to explain with ‘heat’ analogy.
Imagine a piece of pipe with water flowing through it. The friction will cause the pipe temp to go up. This is equivalent to a resistor connected in an electrical circuit, AC or DC. Now this pipe don’t care as to which direction water is flowing inside. The water can be flowing in one direction constantly and create X amount of heat. (Equivalent of DC). Moving the water back and forth through the pipe can create the same effect, which is equivalent of AC. (Side note: The AC value which creates the same effect in a resistor as DC is termed as rms value of AC). It is incidental that in one case it is the mechanical energy and in other it is electrical but both produce the same effect ‘heat’, just a conversion of ‘form’ of the energy! Hail Newton! The water is just a medium; so do not confuse the flow of water with flow of electricity, the energy.
This discussion can go on and on, but I think I did my best to explain. You will do well to remember that ‘energy’ by itself is not visible, it does not have mass, only its effect can be felt or ‘seen’ by change in the characteristics of a visible object.
Also based on its form, it may or may not need a medium to travel. In poor form it requires a medium, such as heat or certain range of frequency and DC in electrical form, but can also travel through space in it much purer from, such as light or electromagnetic waves and cosmic radiations. By Einstein’s theory, which is not contradictory at all, energy in its purest form will have no mass! All of mass m is converted to E then. But as long as we are not converting mass into energy or energy into mass (a feat not achieved yet by man), both can be treated independently.
Material carrying energy is not same as energy! Moving car has kinetic and inertial energy but car is not the energy! Energy does not have a Car, but car does have energyJ.
Let me also clarify the mechanical/hydraulic analogy. Water distribution network and electrical power distribution network is a famous and well established analogy, so much so that the same form of equations could be used to solve either network problem.
There quantity of water equals ‘electrical energy’ only from modeling point of view, the difference being electricity is not a material or mass like water.