Beware! Math ahead!
Here’s my take on the whole thing:
A column of air 1 inch square and as tall as the atmosphere exerts 14.7 lbs of pressure (14.7 psi).
In contrast, a column of water 1 inch square need only be 33 feet high to exert the same pressure. Water is MUCH heavier (duh!).
Weather pressure is usually expressed in millibars. PSI (lbs/square inch) is a more user friendly unit. 1 mbar = 0.014504 psi.
Air pressure during weather changes usually fluctuates from as little as 10 to as many as 60 mbar. If you do the math, this equates (if I did it correctly) to a change of as little as 0.14504 psi to as much as .87 psi.
Think of the water column. To experience a change of 0.14504 to .87 psi (equivalent to weather changes), a fish needs to move vertically up or down 0.7 inches to 5 inches. (calculations excluded, but: change of 1 ft in depth = change of 2.25psi; 1 ft = 12in.)
A change of 0.7 to 5 inches is equivalent to simple wave action in the water.
I don’t know, but to me it seems unlikely that the pressure changes in the weather could have significant influence on fish behavior, when significantly larger changes in pressure occur for the fish simply by moving around in the water column.
Maybe rain on the surface, clouds affecting lighting, etc. could better explain the phenomenon? What do you think?