"Deep breathing" for Buteyko means full lung breathing but not over breathing. Over breathing is filling lungs beyond the ammount needed. The physiological need is to maintain oxygen levels at the same time not affecting your bodily CO2 levels. This is an important point of Buteyko's work: your bodily CO2 levels control your oxygen uptake capacity. Over breathing causes bodily CO2 reduction reducing the bodies capacity to oxygenate.
When Buteyko says "deepbreath" he means do not over breath. This is subject to a proper breathing pattern which Buteyko also defines. Buteyko also developed a simple exercise to evaluate your oxygenation level.
A careful read being aware of this "misunderstanding" will make it all easier to undertand.