Boiler system shows different pressures
The difference between cold and hot pressures is clear: the expansion tank is doing its work, but it only has a finite size, and so as it takes in the expanded water it raises the overall pressure. The tank has a diaphragm across its middle separating the air in the top from the water in the bottom. As the water expands, it pushes the diaphragm up, but that compresses the air and so increases the system pressure. The larger the tank the less the pressure will increase for a given amount of water displacement, but as this isn't a high-precision situation it doesn't really matter if the pressure changes somewhat, so nobody buys a 50 gallon expansion tank.

On the two pressure meters displaying different values, my guess is that they aren't very accurate, and so one is reading high and the other is reading low. Interesting note: your expansion tank meter displays a 25% increase in pressure from cold to hot, which is darn close to the 22% increase in pressure shown by the boiler meter. This implies that they're off by a scale factor, but they'll both read zero PSI when there's no pressure.