>> which case it shuts itself off until the voltage comes back up again.
>
> I always thought they were just wired up to work in the "on" position but 
> not the "start" position.

Before cars turned into computers, the radio system typically was just 
connected directly to the battery through a fuse, and has an "ignition" 
signal line to indicate whether it should be on or not (this signal is on in 
both the "on" and "start" position, obviously).

Car manufacturers give all sorts of detailed specs for radio systems, 
detailing exact input voltage patterns and what the system must or must not 
do.  For older radio systems they only had to work down to 10 V or 
something, and when the voltage dropped below that they had to behave 
sensibly (ie not blurt out massive clicking noises or static).  This was 
purely because it is cheap and simple to make a radio work with a supply 
between 10-16V compared to 0-16V (yes, newer radios do have to theoretically 
work with a 0 volt supply for a fraction of a second).

Recently though, especially with these systems where the engine is switched 
off every time you stop, the radio system is required to keep working during 
engine start events.  There are quite detailed specs about what the radio 
must "survive" in terms of supply voltage fluctuation, because the supply 
voltage is anything but smooth in a car.

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