Effect of single-potentiometer impedance

I am having trouble understanding the effect of potentiometer impedance when the pot is used e.g. to input a value via analogRead(). My question is, does the pot's impedance value actually matter? It's acting as a voltage divider, and so the voltage going out the wiper is entirely relative to the position of the pot, not its absolute value. E.g. pot at 50% travel, Vout = 50% for any pot impedance value.

What I can't understand is this: the pedal for my welder has a 22k pot in it. Other pedals, for other welders have 47k pots in them. And people seem to care very much about getting the right pot value, so it must matter. But when I hook a pot up to my Arduino, it doesn't seem to matter whether it's 10k or 100k--I get an AnalogRead value from 0-1023 regardless. So why does it matter?

I understand that when two resistors are wired in parallel, then things get more complicated. Maybe something like that is happening in the welder? In other words, maybe the pedal pot is actually in parallel with a resistor that's inside the welder. Obviously, y'all aren't psychic about the internals of my welder, but the situation where you have a potentiometer that is used to input a value, and the pot must be a certain impedance seems generally common enough that maybe someone will have some insight.

You are correct about the basic Arduino configuration. The pot is connected between 5V and ground, and the wiper is feeding an essentially infinite resistance.

In other cases the "surrounding circuitry" can make a BIG difference! (A parallel resistor across the pot in the typical Arduino circuit (from 5V to ground) won't do anything.)

There are lots of applications were a regular resistor (or capacitor) value is not critical at all and other cases where precision is very important. Part of engineering is knowing what's critical, and what's not.

And people seem to care very much about getting the right pot value,

I don't know about the welder but besides getting the right value, sometimes a logarithmic or "audio taper" pot is used (where the physical center is not 50% electrically).

P.S.
Before changing the pot in your welder pedal, try some [u]Potiometer Cleaner[/u]. Spray some into the pot and rotate it back-and-forth rapidly several times.