On average, events X and Y are positively correlated if they usually occur together even though occasionally they do not. This lack of perfect correlation is due to A) the natural variation of other (less important) causal factors, or B) imprecise measurement of their values. A and B are also known as 'noise'.
All causation implies temporal separation -- causal event X occurs before caused event Y. The trick is to identify which occurred first AND changed the frequency of the second.
An example is the assertion: "The presence of rain causes people to carry an umbrella". Of course, people carry umbrellas even when it doesn't rain, or don't carry umbrellas when it does rain, but on average, on a day when more people carry umbrellas than usual, it's usually a rainy day. The scientific question is: does people carrying umbrellas cause rain? Or does rain cause people to carry umbrellas?
If the natural variation of rain occurs in some detectable manner (e.g. light rain vs heavy rain) and you see direct variation in how people carry umbrellas (less rain thus fewer umbrellas), then it's more likely that rain causes umbrellas because rain variation correlates positively with umbrella variation. This is effectively confirmed if on several days you see that more people are carrying umbrellas than usual but it's NOT raining harder, then probably carrying of umbrellas does not cause it to rain. (Maybe umbrellas were being given away for free on that day, or the weather forecast threatened more rain than actually arrived, causing more umbrellas to be carried.)
Thus when rain amount rises or falls (due to natural variation or noise), you should see the amount of umbrella carrying follow accordingly. However if the reverse relationship occurs less often or not at all, this implies that rain does indeed cause umbrellas, and not the reverse.
If you disregard noise/variation as an indicator of which event is cause or effect, then neither of the events you propose is clearly the cause of the other. Because variation in barometer pressure is likely to be perfectly correlated with variation in storms, there's no noise/variation in either event that isn't also present in the other, so neither emerges as more likely to be the cause of the other.
This strategy of identifying the causal event works only for pairs of positively correlated events whose variations/noise sometimes do not occur together, like an increase in umbrellas without an increase in rain.
Can barometric pressure rise or fall due to causes other than storms? Can storms arise without being caused by a rise in pressure? I'd say maybe yes to the first (an elevation change of the meter, or a storm front that passes you very quickly but whose clouds don't pass directly overhead — maybe). But I'd say definite no to the second. If you are hit with rain from a storm, your baro pressure will drop. Thus storms cause pressure to drop, but pressure drop does not cause storms.
All causation implies temporal separation -- causal event X occurs before caused event Y. The trick is to identify which occurred first AND changed the frequency of the second.
An example is the assertion: "The presence of rain causes people to carry an umbrella". Of course, people carry umbrellas even when it doesn't rain, or don't carry umbrellas when it does rain, but on average, on a day when more people carry umbrellas than usual, it's usually a rainy day. The scientific question is: does people carrying umbrellas cause rain? Or does rain cause people to carry umbrellas?
If the natural variation of rain occurs in some detectable manner (e.g. light rain vs heavy rain) and you see direct variation in how people carry umbrellas (less rain thus fewer umbrellas), then it's more likely that rain causes umbrellas because rain variation correlates positively with umbrella variation. This is effectively confirmed if on several days you see that more people are carrying umbrellas than usual but it's NOT raining harder, then probably carrying of umbrellas does not cause it to rain. (Maybe umbrellas were being given away for free on that day, or the weather forecast threatened more rain than actually arrived, causing more umbrellas to be carried.)
Thus when rain amount rises or falls (due to natural variation or noise), you should see the amount of umbrella carrying follow accordingly. However if the reverse relationship occurs less often or not at all, this implies that rain does indeed cause umbrellas, and not the reverse.