I think you have it figured out. When you see a reference to a bite of air in Total Immersion, it is a first step in the progression to fully functional breathing while swimming. Since breathing has the potential to compromise the swim stroke, is is helpful just go for a 'bite' of air when you first add breathing to your solid swim stroke. When you've added breathing to the swim drill or stroke, you want to be very conscious that the swim stroke is not changed to accommodate the breathing. Once you can drill or swim and take a bite of air without problem, you can progress to deeper or normal breathing.
I am also a fan of Chi Running. I think the discussion you reference from Chi Running fully applies to swimming as well as running.