I'm curious how it matters, can you be more specific? You didn't answer the question, what is the "correct" way to breath? What bad things happen if you don't breathe correctly, aside from passing out or dying? You'd have to be consciously trying to not breathe for that to happen, right? If there's discomfort, what adjustments am I making naturally that might be "incorrect"? What about breathing is so non-obvious that it requires training and not instinct?
I run a lot, and I normally don't have to think about breathing. If I need more air, I breathe faster or deeper without thinking about it. My cadence very naturally goes at a multiple of my stride, because it's more comfortable. I breathe through my nose sometimes and through my mouth sometimes. The deciding factor is usually whether my mouth is dry or there's a lot of dust in the air. I rarely notice which. Both my nose and mouth are holes in my face that facilitate getting air into my lungs, so I'm not getting why there's something "correct" about using one or the other?
What does this notion that there's a "correct" way to breathe have to say about normal variations in human physiology, e.g. larger/smaller lungs, skinny vs heavy, larger/smaller sinuses, mild asthma, etc.?
As you mention, it's about rhythm and timing with your stride to maximize comfort and minimize effort so that running is challenging but not unnecessarily strenuous.
Being conscious of your breath forces you to manage pace. If you're trying to maintain an aerobic pace, noticing when you start breathing hard is a signal to dial back.
Additionally, particularly in colder climates, it's also helpful to breathe in through the nose and out through the mouth, in order to warm and moisten the air before it hits your lungs, thus increasing comfort.
Sounds like you've probably already landed on an effect breathing technique for you. But just because you figured that out on your own, don't assume everyone else naturally will. People unfamiliar with aerobic exercise may not realize it's abnormal to be gasping and out of breath while running.
I run a lot, and I normally don't have to think about breathing. If I need more air, I breathe faster or deeper without thinking about it. My cadence very naturally goes at a multiple of my stride, because it's more comfortable. I breathe through my nose sometimes and through my mouth sometimes. The deciding factor is usually whether my mouth is dry or there's a lot of dust in the air. I rarely notice which. Both my nose and mouth are holes in my face that facilitate getting air into my lungs, so I'm not getting why there's something "correct" about using one or the other?
What does this notion that there's a "correct" way to breathe have to say about normal variations in human physiology, e.g. larger/smaller lungs, skinny vs heavy, larger/smaller sinuses, mild asthma, etc.?