Electronic – Why does a larger wire cross section mean less atom-electron collisions when there must be more atoms, too

The wire gauge you need is a function of several things.

• The acceptable voltage drop or power loss (that appears to be the only thing considered in the website you linked). The voltage drop (and power loss) is proportional to wire length and inversely proportional to the cross-sectional area of the wire- in other words inversely proportional to the square of the wire diameter (assuming constant current).
• The acceptable temperature rise. This is a function of the number of current-carrying wires bundled together, the environment (maximum ambient temperature and air pressure or altitude, for example), the insulation type, the wire type (some types of wire are plated to withstand higher temperatures than bare copper without corroding).
• Regulatory requirements and other considerations- for example, the wire may be rated for 200°C insulation, but you might not want the wire to run that hot.
• Fusing- the fuse or circuit breaker should protect the wire in the case of faults such as overload or short circuit.

Very short lengths of wire can depend on heat sinking through the ends (indeed, in a vacuum, that may be the main heat loss mechanism), but usually that's not taken into account.

Normally you'd run through a checklist such as the above to make sure ALL the requirements are satisfied simultaneously, so you might find that using PTFE insulated wire allows you to use AWG 18 wire, but because of the voltage drop limitation you'll have to use AWG 12 wire.

Half the point of using voltage, current, and resistance is that we don't have to care about what the power supply and resistor are made of. Whether you're using eight 1.5 volt AA batteries in series or one 12 volt car battery, if you connect 150 ohms across the terminals you will get about 0.08 amps. (The batteries have some internal resistance, but it's very small.)

It might help to look at this from a physics perspective. You have an electric field that pushes electric charges around, giving them energy in the process. That energy is then lost as the charges move through a medium. (More specifically, moving electrons collide with atoms.) The rate at which the charges move depends on both the strength of the electric field and the medium's ability to let the charges move around unimpeded. It turns out that the medium can often be described by a single parameter called the conductivity. This gives a simple relationship, known as Ohm's Law:

$$\vec{J} = \sigma \vec{E}$$

where \$\vec{E}\$ is the electric field strength and \$\sigma\$ is the conductivity. \$\vec{J}\$ is called the current density, and represents the rate of charge flow. This is a microscopic relationship. Note that the current and E-field are vector quantities. Doing physics with 3D vectors is a lot of work, but fortunately we have a simpler option -- circuit theory. In circuit theory, we talk about voltage (energy) instead of electric field strength (force). Just as in basic mechanics, this lets us deal with complicated situations in simple ways.

To make the jump to circuit theory, we have to change to using macroscopic variables. Instead of talking about the electric field strength at every point, we talk about the energy difference between two points. Instead of talking about the conductivity of a medium, we talk about the conductance of a physical object. Instead of talking about the density of charge flow, we talk about the total current through a wire. Now we can use Ohm's Law in its macroscopic form:

$$I = GV$$

where V is the voltage (energy per unit charge), G is the conductance (measured in amps/volt, aka siemens), and I is the current.

You can think of the voltage as being a sort of pressure that pushes charge through a circuit. The conductance tells you how much current you get for a given voltage. Now you're probably thinking "Where the heck did conductance come from? I was asking about resistance!" Well, resistance is just the reciprocal of conductance:

$$R = \frac{1}{G}$$

Resistance is more convenient because we're usually more interested in low-conductance elements in a circuit. With resistance, we can use a normal number like 150 ohms instead of a tiny fraction like 0.00667 siemens. This gives us the most familiar form of Ohm's Law:

$$V = IR$$

Now, back to your question. The reason that one car battery and eight AA batteries have the same effect on your circuit is that both of them are producing similar electric fields. Here's an analogy -- pulling a wagon over a rough surface. It takes a certain amount of force to make the wagon roll at a certain speed. It doesn't matter whether that force comes from your hand or a car engine -- it's still the same force.