Is there a proper way of breathing?

No, I don’t believe there is.

There is a correlation between the breathing pattern and your state of mind. When you’re anxious you breathe one way, when you’re relaxed you breathe another.

There’s a case to be made for the wisdom of the body. For the most part, it breathes the way it needs to breathe. Where meditation is concerned, this means that the breath adjusts automatically in response to the mental settling that occurs as you meditate. You don’t need to make this happen.

Suppose you’re meditating and you feel anxious and you want this to go away. So you begin to consciously manipulate your breath, slowing it down and getting it down into your belly, like you’re “supposed” to do. But the fact is that you’re anxious, and your body needs to breathe in a particular way when it’s anxious. Try to breathe otherwise and you’ll tend to feel like you’re not getting enough air. This will actually ratchet up your anxiety, and it turns into a fight between your will to impose a certain way of breathing and your body’s natural instincts.

Now suppose again that you’re meditating and you feel anxious. But rather than wanting that anxiety to go away, you simply accept that this is just the way you happen to feel at the moment, and that’s OK. You put your attention on your breath. It’s fast, shallow and chesty, because that’s how your body needs to breathe when it’s anxious. So that’s OK, too.

You keep your attention on the breath, without any effort to control, just observing how it is moment by moment. This is itself relaxing. Gradually, your breath will begin to respond to this relaxation, slowing down and settling into the belly. And this all happens without your trying to make it so.

Having an idea about how you ought to be breathing is entirely unnecessary.

Well, as long as you aren’t passing out, I’m guessing you are doing alright.

Not through your mouth.

Its usually recommend you breathe through the nose.

In, then out.

Our whole culture has some problematic thought patterns, such as reducing situations to statements that are right or wrong, or measures which are good or bad, or normal or abnormal, or whatever.

About breathing, I think it would be more accurate to say, that breathing is one of the most fundamental physical things we do, and that like all physical actions, people tend to learn how to do it, and stop paying attention to how we do it, after we think we’ve learned how. This leads to habits and limits that we don’t generally even realize we have, and which limit what we can do. Eventually, these can lead to problems or at least reduced possibilities. Breathing also affects how much oxygen our brain and body get, and our nervous system and other body systems in general. It affects our tension and state of mind, and our health.

There is not one right way to breathe. However it is very helpful to be aware of our breathing, and to be able to explore the range of ways we can breathe, and to be able to adjust it. There are various techniques (apart form just using our innate natural abilities, if we haven’t too deeply buried them in our “adult” thought/unconsciousness patterns): e.g. meditation (q.v. prana breathing), Breathing can be directed and used to relax our bodies and minds.

I’ve been breathing more or less successfully for 30 years. I think I’m ok.

I do take deliberate deep breaths to calm myself down. It seems to work alongside a deliberate quietening of my mind

Shallow breathing doe not remove enough carbon-dioxide. The more oxygen that reaches the brain the better.

In yoga practice, there are lots of different ways of breathing. Thinking about the breath and noticing the way that you breathe can be very beneficial. The way that you described concentrating to make the inhale and exhale equal in length (and training yourself to make them longer, optimally six counts) is stress reducing. Other methods of breathing have other goals. “Here”: is just one more, which is very interesting.

Actually yes, as @Bill1939 mentions. Survival breathing is just taking the usual, unconscious, shallow breaths, but deep abdominal breathing oxygenates the body better and leads to deeper relaxation such as in meditative practices.