The water analogy
The water analogy does not go as far as you expect it. You should only consider that pressure differences relate to voltage differences and flow rates to currents.
It is not meant to have a perfect equivalence, but it is a great way to help visualising or starting to get a feel of what a voltage is, what current is, and what resistors, etc. are.
When the difference in pressure is higher, the flow rate is higher. The same way how higher voltage results in higher current.
There are other differences. Water will have higher flow in the center of a tube, while current will be higher on the outer edge (skin effect). So the analogy is only a way to get started.
Speed of electrons
The speed that really matters is the signal speed which is about half the speed of light in a cable or in a PCB. And signal speed is actually related to displacement of power.
So what seems strange is that electrons are so much slower.
So let's look at the water analogy.
Suppose that I have a tube with a section of 1 dm^2, which means that there is 1L in 10 cm of the tube. In order to receive 1 L, the water in the tube has to make a displacement of 10 cm.
Suppose that I then have a bigger tube, which has a section of 1m^2. In that case, to receive 1 L, I need a displacement of only 1 mm.
It is very easy with the tube of 1m^2 to have a fast fluctuation of 0 L/s to 1 L/s with very small displacements. In the small tube the water would need to move 100 times faster.
Electrons in a wire work more or less the same way - there are quite a lot of them. In fact for 1 amps in a wire with a radius of 2 mm the speed that the electrons must exhibit is only only 23μm/s (source https://en.wikipedia.org/wiki/Drift_velocity#Numerical_example).
So similary the signal can change from 0A (Coulomb/s) to 1A (Coulomb/s) very quickly with these "very low" speeds of electrons, the same way we can change from 0L/s to 1L/s in the water analogy.
Electromagnetic waves around a wire are determined by the current fluctuations in the wire. And as you can see, the speed of these fluctuations can be very high with only low displacement speeds of the electrons.
Voltage is what pushes the electrons...
Voltage is defined as the "work" (joule) needed to displace one unit of charge (coulomb), which itself is a number of electrons.
In an electrical field, we can identify a voltage level at a given position that indicate how much work is needed to move a charge from 0 V to that position.
These voltage levels represent the electrical field, which is an "invisible" force acting on electrons, in a analoguous way as gravity acts on water.
I prefer to think that electrons are drawn to the lower energy level, but as they loose energy they do flow to the positive potential by convention (so the higher voltage potential is in fact the lower energy level, but forget that as it makes things more awkward).
That should explain the role of "Voltage" - Voltage is a measure in the electrical field.
Why is voltage lower at the negative terminal? By convention: it could have been chosen the opposite way and the current negative terminal could have been the positive terminal. The negative terminal is called "negative" because the voltage is lower there, and that is a choice.
Analogy with springs
Comparing to strings is another ball game, and surely not the best one as springs do not seem to have something that can compare to current.
Voltage could be somewhat compared to the force to keep the springs tight in the box. But there is not an "infinite" reservoir of what could represent electrons and current.
Analogy with fuel
This is even more difficult than the analogy with springs. There is a chemical reaction that results in heat and pressure that is transformed in a displacement.
The fuel is displaced to the combustion engine by succion. The tank itself does not represent the voltage.
The voltage level in the fuel is somehow in the chemical bonds. These bonds are stable enough until enough heat is applied to break them. I can't think of a proper analogy in electricity for that. I could find one with water though.
? Voltage is the force that pushes electroncs apart ?
No, from what I indicated above, voltage indicates the work that that was need to move the electrons to that voltage.
Pushing or attraction is a question of a point of view.
Update 1 (in your question)
This is true up to a certain point - electromagnetic fields will not help keeping that true.
Remember when you have a metal tip, electrons will tend to all move towards the tip which seems difficult to explain when considering that electrons want to be distant from each other.
The reality is really a lot more complex and we are working with analogies all the time.
Even mathematics could be viewed as some kind of analogy. We can do calculations about signals "themselves" or use Fourrier, z, and other transformations to compute them.
As said, there is the question a point of view.
Electrons are attracted by the holes or pushed away by other electrons.
Fact is that they can not really go anywhere if there is no hole. Electrons are attracted by the protons and any unmatched proton creates a hole. Electrons can change energy levels within the same atom and emit or absorb a photon when they do so. The scientific world has not been capable of combining the theory about photons with electromagnetic waves.
Electrons can also be entangled, which allows for teleportation. When an entangled electron absorbs a photon, the other one emits one. That is displacement of energy and hence mass without an electrical path.
So in the end, we choose the analogy that fits best to solve a given problem, and we sometimes use more than one to be confident that the problem is solved.
Electronics (as water) will tend to move in a position where they have lower potential energy.
Suppose that 1 L represents 1 J of energy. If you move 1 L horizontally, you displace 1J. If you move \$1 m^3\$ over the same distance you displace 1000J.