What really is voltage, what effect does a higher voltage have on an unit of charge and what makes electrons flow?
The trick is not to use bad analogies. As you've noticed, the water analogy is bad, because "pressure" is an inherent property of a volume of material, and not something that only exist in relation to some reference pressure¹.

Instead, try to simply go by the plain definition of what voltage is, and things will get easier.

Your fuel analogy is another bad analogy, because the electrons don't leave your circuit. Your fuel does. Honestly, the model we use for basic electronics isn't that hard, analogies just make it harder.

Voltage is the amount of energy per charge. That's it!

Wait. Energy per charge? Energy – how? A charge itself doesn't work, does it? 

You're right, a charge (say, a 5V-charged metal sphere floating in outer space) don't exert a force on anything or so, they don't heat up the sphere by themselves, they don't shine light... 

But: if you have a second sphere with the same charge, but negative voltage, these two spheres will attract: a force is exerted on both, and that force is proportional to the voltage.

Same happens with your battery: the chemistry inside leads to a potential on the positive pole that is (say, 1.5V) more positive than on the negative pole.

That means that 1 Coulomb (that's the unit of charge, it's 1 C = 1As, the amount of charge that flows through a cable when you draw 1 ampere for 1 second) charge has 1.5 V · 1 As = 1.5 VAs = 1.5 Ws = 1.5 J energy. Done!


  To me, it seems that voltage is simply the force that pushes electrons apart from each other. 


pretty much, yeah.

You can think of in electrons, but you'll later learn that thinking in electrons doesn't have any more physical relevance than thinking in "missing electrons", so just think of charges, then you can say


  it seems that voltage is simply the proportionality between the force that pushes charges apart or attracts them to each and the amount of charge


or really, as said,


  it seems that voltage is the ability of a charge to exert work


which really is identical to saying


  it seems that voltage is energy per charge.


There we are. You were right!


  electrons flowing from the beginning of the circuit to the end, when in fact the electrons barely more.


Yeeeeah, long story short, making good animations is hard, and especially when you grow up with the water analogy, it's even harder.

In reality it's really more like a "potential slope". At the top of your battery, relative to the bottom / ground connection, you have a high potential. Your drawn connection is an ideal conductor, i.e. no voltage drops across it, so at the top side of your resistor, you still have +1.5V relative to the bottom side. Now, along the length of the resistor, the electrons have to logically have less and less potential – and that's exactly what happens, if you make a graph of "energy per electron" on the y-axis, and "length along the resistor" on the x-axis, it's just going to be a falling straight line from y=1.5 V at x=0 to y=0 V at x=end.

So, I think you're righter in understanding electricity than these analogies are. 

Generally, analogies are rarely more useful than just as an "introduction". If something is complex in reality, then the analogous thing you use to explain it needs to be complex, too. So, what's the point of using an analogy if it's as complicated as the thing you actually want to understand?

Luckily, voltage isn't that complicated.  (If you think about water circulating in pipes – hell, that is complicated. Fluid dynamics really is complicated, and also, quite different from how electricity behaves.)


¹ I really don't know why the water analogy is so commonly used. It's not really a great analogy – whenever people use it, they have to define what the water in their model does in ways that water, pipes, volumes, pumps don't really work in reality. I could rant about this for hours – but it wouldn't help you.