I love this quote from the article:
Is it the "cabin altitude" or the "pressure" that is higher on shorter flights? "Cabin Altitude" and "pressure" have an inverse relationship. As "cabin altitude" increases, the cabin "air pressure" decreases.

Aircraft cabin pressure systems normaly operate as a fixed maximum differential between inside and outside pressure. This maximum differential is a defined value for each hull type and is part of the aircraft's certification.

The aircraft cabin pressure systems will normally be operated in an automatic mode which holds the inside air pressure constant until the max differential (or the pre-set differential) is reached, after which the inside air pressure drops at the same rate as the outside air pressure, maintaining a constant differential.

It is this cabin air pressure differential that restricts the operating ceiling of an aircraft. High altitude aircraft such as Concorde must be certified for a higher differential between cabin pressure and outside air pressure.

"Cabin Altitude" is defined as the altitude at which the current cabin air pressure would be experienced if the aircraft was not pressurised. It generally measured in feet and would normally be in the 6000-8000 feet range. Note that the North Rim of the Grand Canyon is at 8000 feet, so a cabin altitude of 8000 feet is just like breathing at the North Rim.

Cabin Altitude can also be adjusted by the crew within the limits of the maximum allowed differential. If they want a higher cabin pressure (the same as lowering the cabin altitude), then less air is allowed out through the out-flow valves. If the max differential has been reached, then the only way to increase the cabin pressure is to reduce altitude.

So the cabin air pressure is completely reliant on the altitude of the flight and the max differential for the aircraft type. I tried to locate info on the max pressure differential for A320 vs B737 but oculd not find it.