"Pressure" as used by a meteorologist means the weight of the air above the surface. The classical way of measuring it involves supporting the weight of a column of mercury by the pressure generated by the weight of the column of air above it, typically about 30 inches or 76 cm of mercury. Yes, a column of colder air would be more dense, and so weigh more than a column of warmer air, all else being equal. The other factor that has to be considered is the height of the air column - a taller column with the same temperature air would exert more pressure at the ground than a shorter one. Of course, in the atmosphere we don't have a nicely well defined height to the air column - it just keeps getting less dense as you go up. But you can think of the air column as having an equivalent height. Think of a water tank. Imagine that the water is well mixed so everything is the same temperature. If the water is at rest the pressure at the bottom of the tank will be the same everywhere. But if you slosh water around in the tank and create waves, you find the pressure at a point at the bottom of the tank is greater under the wave crests and becomes less as the crest is replaced by a trough. It's the same idea in the atmosphere. The travelling pressure systems in the atmosphere are very analogous to the tank's waves.
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