Electrical engineer here.
Thank you for a question that is so dear to me, and is so important to our understanding of the universe.
Once you get past the semantics, which are covered in other questions, it seems the core of your question is what is happening when current flows and is it more than the simple analogy of flowing water?
The bulk of this understanding is covered in the laws that Faraday deduced and Maxwell created the formulas for. Maxwell's equations are difficult even for me, I don't usually use first principles like that either; but if you want to spend some time learning calculus and tensors, you too can get down to those basic principles.
I recommend looking at an actual example, such as electron flow through copper wire. A copper wire is made of a lattice (a kind of crystal) of copper ions. Because it is missing some atoms to complete its shell there are free electrons that move through this lattice similar to a gas. You will note if you examine the tables on conductors that some metals that are good electrical conductors also conduct heat well for the same reason - heat is also a vibration of electrons and depends on free electrons to easily transmit heat. There are exceptions of course, such as thermally conductive paste. [https://www.thoughtco.com/examples-of-conductors-and-insulators-608318]
Normally, the free electrons move about randomly in the metal; but when you subject it to electrical potential, the electrons move in a more organized fashion from negative to positive potential. Electrons have a negative charge. They are attracted to the positive end of the battery. The free electrons move through the copper, flowing from the negative to positive terminal of the battery
Note that they flow in the opposite direction to conventional current; this is because they have a negative charge. Most electrical courses use current flowing from positive to negative which is called hole flow, its just a counter example because traditionally it was thought current flowed from positive to negative. I mention this because a lot of web pages and text books will say current flows from positive to negative; its true; if you, instead of looking at the movement of the electrons, you look at the movement of the holes that the electrons leave.
Using the word electrical current is to simply say an electrical charge is moving through the wires due to the action of the potential, say a battery on the wire. Current is the rate at which charge flows past a point on a circuit. The current in a circuit can be determined if the quantity of charge Q passing through a cross section of a wire in a time t can be measured. The current is simply the ratio of the quantity of charge and time. Current = I = Q/t where I is current in amps, Q is charge, and t is time. Q, charge is measured in coulomb. 1 ampere = 1 coulomb / 1 second. That is a bunch of electrons, lets say 1 columb pass a spot on the copper wire in 1 second, then you have one ampere of current.
The path of a typical electron through a wire could be described as a rather chaotic, zigzag path characterized by collisions with fixed atoms. Each collision results in a change in direction of the electron. Because of collisions with atoms in the solid network of the metal conductor, there are two steps backwards for every three steps forward. The overall effect of the countless collisions is that the overall drift speed of an electron in a circuit is abnormally low. A typical drift speed might be 1 meter per hour, with a potential applied!
When you apply an electric potential across the two ends of the circuit, the electron continues to migrate forward. Progress is always made towards the positive terminal. there are many, many charge carriers moving at once throughout the whole length of the circuit. Current is the rate at which charge crosses a point on a circuit. A high current is the result of several coulombs of charge crossing over a cross section of a wire on a circuit. If the charge is densely packed into the wire, then there does not have to be a high speed to have a high current. Instead, there just has to be a lot of them passing through the cross section.
You might think that for this reason electrical current is slow, but as you know its fast, very fast. In fact it travels at nearly the speed of light, and this is proven by Maxwell's equations, yes the nature of what you are asking is described by his four equations.
When you flip a switch it causes an immediate response throughout every part of the circuit, setting charge carriers everywhere in motion in the same net direction. While the actual motion of charge carriers occurs with a slow speed, the signal that informs them to start moving. The electrons that light a bulb do not have to first travel from the switch through the entire length of wire to the filament. Rather, the electrons that light the bulb immediately after the switch is turned to on are the electrons that are present in the filament itself. As electrons leave the filament, new electrons enter it. The electrons are moving together much like the water in the pipes when the pipe is already full of water. The water that comes out of the tap first is the water that was near the nozzle.
Now you ask is it more than that? Well yes, it is. Electromagnetism cannot be separated, electricity is magnetism and magnetism is electricity. Everything that exists, everything you interact with, everything you see, all the radio waves, xrays, the energy from the distant stars, it is all electromagnetic waves and it is governed by the Faraday and Maxwell observations and equations, respectively. What you ask gets to the very fundamental question of mater, energy and existence. I would argue that your simple question is not simple at all, it gets to the core of the fundamental forces of nature.