You need to be careful with analogies. Here are some problems in the analogy you describe:
water starts flowing from a collector (battery)
Nothing in an electric circuit really works like a collector of water. In your analogy, water is electric charge which, in metals, is carried by electrons slowly drifting. Batteries do not store charge, they are not a reservoir of charge (nor of electrons).
Batteries store energy in chemical form. A better analogy is that a primary battery is a coal-fired water pump that will deplete it's store of coal as it pumps water. A secondary battery is a bit like a pump powered by a wind-up spring, it can be run in reverse to wind up the spring. These pumps can only pump water if their outlets are connected to a circuit of pipes that eventually returns to their inlets.
does that mean that the voltage and current increase ...
Voltage isn't something you measure at one point, it's something you measure between two points - it's a difference.
If you measure the voltage at every millimeter of the circuit with respect to the batteries negative terminal you will see the voltage monotonically decreasing as you progress around the circuit‡.
The current measured at any point in the circuit‡ is the same. It neither increases nor decreases
... but the number of electrons would decrease
It isn't very useful to think of the number of electrons increasing or decreasing. Where would they go? Where would they appear from?
You measure a current† of water in litres per second. You measure a current of electricity in coulombs per second (amperes). In a steady-state system, this current is the same in all parts of a simple serial circuit - whether of pipes or of water. A constriction in a pipe cannot make n litres per second of water disappear.
If you slightly turn a gate-valve in a water pipe, the flow of water (litres per second) decreases in all parts of the circuit, including in the pump.
resistance increases causing the pressure and speed of the water to increase but the volume would decrease.
That's not how water works!
If we imagine a simple circuit where a water pump is pumping water around a loop of pipe. The pipe is of uniform size apart from one place where we have a section of narrower pipe.
resistance
The resistance is greater in the narrower pipe (a greater proportion of the water is close to the pipe walls and experiencing friction)
pressure
However the pressure is lower, not higher!
speed
It's the lower pressure that causes the water to accellerate to a higher velocity as it enters the narrow section.
volume
When you say the volume increases, I think you mean the velocity increases. Water is relatively incompressible, it's volume doesn't change much at the pressures applying in our analogy.
The flow rate (volume per second) is unchanged.
Footnotes
† This is one of the areas where the analogy starts to break down. The word "current" is used inconsistently. If you asked someone to measure the current in a river they might give you an answer in metres per second ("current" = average drift velocity of H2O molecules) instead of litres per second ("flow" = litres per second passing a fixed point).
‡ This answer applies only to a simple circuit of battery and resistor connected by copper wires.
You said yourself voltage is like pressure and current like flow rate. If you measure the pressure in your water line just before the sink valve, it will be maximum when the valve is off, and go down as the valve is opened more (letting more water flow). The pipes back to wherever the water pressure is maintained have some resistance. This resistance times the flow causes a pressure drop.
In the case of the processor and its power supply, the power supply is more like a pump. This is like a water pump that has a maximum flow it can sustain. If you tried to draw 10 gal/min from a 5 gal/min water pump, you're not going to get a lot of pressure, and you're not going to get 10 gal/min.
It's the same thing with a processor and its power supply. Let's say the power supply is rated for 3.3 V and 200 mA. That means it only promises to maintain 3.3 V if you draw 200 mA or less. If you try to draw more, various things can happen, but they all include dropping the voltage.
Note that the section you quote talks about the processor malfunctioning when it gets too low a voltage. This is not the same as getting damaged. Low voltage to a processor is not going to damage it, but you can't expect a processor designed to run on 3.0 to 3.6 V to work at 2.0 V. At too low a voltage, transistors can't be turned on fully anymore, which means they can't turn on other transistors fully, etc. The digital levels become undetermined and take longer to propagate thru the logic. The processor performs meaningless operations on meaningless values, if it can "operate" at all. At some points the clocks fail to clock.
Best Answer
First, analogies can be good for getting the basic concepts, but can only go so far. A common pitfal with the water analogy is that, unlike the air around a hose, the air around a wire is non-conduting. It can also make the requirement for closed loop current hard to grasp later. Someone holding the end of a hose and the water spilling onto the ground is a mental picture you want them NOT to have of electric circuits.
This doesn't mean the water analogy can't be useful. It can, but don't make too big a deal of it. Use it to introduce the concepts of voltage (pressure) and current (flow rate), but don't go too far with it. The details don't match well with electric circuits, so get off the water analogy once the basic concepts have been introduced.
NO! It seems you have yourself fallen into the water analogy too deeply. Analogies are aids in learning, not substitutes for actually knowing something.
There is no simple electric equivalent of a open-ended hose. If the hose were to continue, the pressure after the constriction would be lower than in front of it. Actually with the open-ended hose it is even lower. You can't observe pressure, only flow. You see a jet of water and erroneously assume it has high pressure. It may have high velocity, but that's not pressure.
This is yet another source of confusion from the water analogy. Water velocity (meters/second, for example) doesn't really map to anything useful in electricity. Stick to flow rate (gallons/minute), which maps to electric current flow (Amperes).
One of the big problems with the water analogy is that we intuitively know that water has momentum, even when we aren't aware we are looking at water that way. The jet of water produced by the nozzle relies on momentum. There is nothing analagous in electricity (yes, I know about electron guns, but by the time you understand those, you're way past the initial crutch of the water analogy).