Interesting, let us know if it stays fixed.

I can see how a failed battery could lower voltage at idle and cause faulty pressure switch readings. But it's also possible it's intermittent, and disconnecting the old battery cleared a code.

 

Yes, the battery is only used to start the car. Once the engine is running, all the electrical needs are supplied by the alternator (except MAYBE in super rare circumstance when all the electrical power consuming devices are on and the total load exceed's alternator's capacity, but I doubt non-modified car would have that possibly though).