As a trumpet teacher the issues and confusions about breathing come up all the time.
I have found a good way around them that I would like to share.
We can probably all agree that to play long phrases we need quantity and to play high and loud we need air speed and power.Regardless of which muscles are used it is the result that is important and this can be measured, and improved in the following way.
Buy one of those 4 litre boxes of cheap wine with the silvered plastic bag inside. When it is finished remove the pouring plug, wash out the bag, drill a hole in the plug large enough to take your mouthpiece shank. Then practise breathing, filling up the bag via your mouthpiece, while watching a clock on the wall. The bag has a capacity of about 6 litres which is a large mans maximum lung capacity. (The two litre wine cask bags have a 3 litre capacity empty and are excellent for children)
At first you will probably find it takes 2 or 3 inhalations and about 30 sec to fill the bag. With practise you will be able to do it in 5 sec and from one breath. Forget all this crap about which muscles to use just practise with this simple device and your body will work out the best way.
Happy bag blowing........
Regards Peter Kartu, Sydney Australia.