lets see if we can muddy the waters even more. I will try to see if I can explain things, while sticking with the previously established water theme.
While current is measured in amps, and voltage is "pressure", amps would be more analogous to volume, more so than flow. flow would be a measure of resistance, which according to Ohm, would be would constant in a given medium. however the 2 have a direct correlation. That is however putting the cart before the horse, so we should start at the beginning with some basic laws and definitions.
electricity can be broken down (mostly) into three fundamental and separate aspects, and can be expressed in the following formula
V*A=W
Where V = volts, A = amps, and W = watts
in keeping with the water theme, volts would be the pressure, Amps would be the volume, and watts would be the potential for work (sorry no real water analogy here), which can be compared to HP. The more HP you have, the more potential for the work that can be performed.
The hose analogy from earlier is still valid, with a few caveats. lets assume you have a 100 gallon water tank, and the same said 2 hoses. if you try to pass 100 gallons per hour, through the garden hose, you will encounter more resistance than you would with the fire hose, in fact you could get to a point where you are passing too much water through in a given time and cause the hose to burst. The same thing happens in wiring, but for different reasons.
The reason that the thicker (smaller gauge) wire could produce the 300 amps, is simply because it does not have the resistance that the thinner (bigger gauge) wire does, before failure. Again, referring back to Ohm, resistance is a function of the medium involved, so copper, being less restrictive than say aluminum, could carry more amperage, in the same gauge wire. An Ohm, is a unit of resistance in a conductor. All conductive mediums have different resistance, as do different gauges within the same medium. That is, a 6awg wire can handle more current than a 10awg wire can. the flip side is, that a 6awg also inherently has more resistance, so trying to push low currents through is not going to happen any more than high current on too small a wire, though for different reasons.
too small a current on too big a wire, is like trying to pull a tractor trailer uphill with a goat. Sure, the goat may make it for a little while, but eventually he will just wear out and never make it to the top.
Too big a current with too small a wire is like trying to pass a baseball through a garden hose. It might make it a little bit, but sooner rather than later the hose is going to split.
so, back to V*A=W, and Ohms law, which states that the current through a conductor is directly proportional to the volts (Potential Difference) which can be expressed: I=V/R, where I = amps, V = volts, and R =resistance, we can work out the resistance of a known gauge wire, and calculate the wattage of the appliance or fixture we are trying to power, with our given current and voltage. or in other words, we know we have a 12v system, with a 40 watt bulb, we can use some algebra from our formula V*A=W or W/V=A. 40(w)/12(v)=3.33(a). Then on to Ohm I=V/R or V/I=R. 12(v)/3.33(a)=3.6, so we would size our wire accordingly.
so, back to Sean's pressure/volume difference, when reading current, you are reading the amps that are travelling across the conductor, are you measuring pressure or volume? The answer is yes.

You may start off with the same volume (amps) upstream, but because of the resistance found in different conductors (big vs small, dissimilar conductors) the pressure downstream can, and will be different. The inverse is also true, you can start with the same voltage upstream, but end up with different current downstream, for the same reasons.
Batteries.
electricity is conducted 2 ways (mostly), through electrons, and through ions. when conducted through wire, electricity is transferred through electrons, this is partly where Ohms law, and thermodynamics come in to play with regards to wire sizes. a By product of transferring electricity through electrons (that little elementary particle that zips around a proton) is heat, excessive amounts of heat. Trying to push too high a current through too small a wire will produce more heat than the wire can dissipate, causing failure.
electricity in batteries is transferred via ions in an electrolytic solution. a battery is comprised of an array of cells. cells are (generally) 2 dissimilar materials, with an imbalance of electrons separated by an electrolyte. the imbalance of electrons will always try to right itself, and get as close to equal as possible (more or less), 1 electron to 1 proton. the ions in the electrolyte are imbalanced and "transfer" their energy from one pole to another (cathode and anode) and spit out the excess electron at one end (electricity). [This is not exactly how it happens, but it is close enough for this discussion]
The more cells (array of cells) a battery has, the more power (amps) that battery has available to it. What (mostly) determines (not always) the voltage available to a battery, is the chemicals used, and the dissimilar mediums. the interaction, and exchange of electrons within, and between the mediums will produce a certain voltage.
to get back to Sean, and why are some 12v different? that is the number of cells a battery has (mostly).
how do we conceptualize them?
try this analogy. in a room, you have 5 thirty year olds. In another room, you have 2 twenty nine year olds, 2 thirty one year olds, and 1 thirty year old. no matter how you slice it, the average age is thirty, in either room. thirty would be the volts (12v). and 5 would be the amps. try as they might, they can neither make themselves a year older, nor make their total more.
now in another room, you have 15 thirty year olds. they are still all thirty (12v) but now they have tripled their quantity (amps)
the same holds true for a battery. you can add more array, but not necessarily increase the voltage (series vs. parallel)
you understand how he tested for amps at the device. you know that he tested for current, a measure of amps, and you know he went from a 6v system, to a 12v system, doubling his previous voltage. so you can use your formula for I=V/R, insert known parameters (v and amps) V/I=R or 12.6/.076=165.78, or 165(166) ohms resistance. since he doubled his voltage, and we want to half it for the fuel gauge (6v), we need to double our resistance, or our Ohms. We already know, from our equation, that we have 165 ohms, and we are looking to double the total resistance to 330 Ohms. 165 are already present, so we need to add 165 Ohm resistor to get a total of 330 Ohms. Basically you are using algebra to solve for 1/2 V (to reduce 12v back down to 6v). V=RI or R=V/I or I=V/R
Hopefully I have succeeded in providing more questions than answers, and you will never get a firm grasp of electricity
just like I promised, clear as mud.