Pilates Breathing vs. Mindfulness Breathing

I have written about both Pilates and Mindfulness lately, as both are areas I have been pursuing in my quest for optimal health.  I discovered a dilemma between them the other day.  It concerns how I breathe, an activity that I am quite fond of, and plan to continue to do until the moment I die.  I have been reading the book Full Catastrophe Living by Jon Kabat-Zinn, one of the big names in mindfulness.  One of the basic ways of meditating involves focusing on your breath. Kabat-Zinn recommends diaphragmatic breathing, or belly breathing.  In this type of breathing, you relax the belly, letting it rise and fall with each breath.  He states that this type of breathing allows the lungs to expand more and to take in more air with each breath, and is slower and deeper than chest breathing. 
In Pilates, the focus is on strengthening the core muscles, mainly the abdominal muscles.   To do this, we are instructed to 'pull the belly to spine', and to 'keep the abdominal muscles engaged' (or pulled in) throughout the exercises.  Breathing is an important part of Pilates, with specific breathing patterns. Keeping the abdominal muscles pulled in while doing the exercises can be a challenge at times.  Strong abdominal muscles support the torso and organs, decrease strain on the back, and provide structure from which the whole body can function better. 
When I am doing my water walking exercises, I often remind myself to pull in my belly, as I do in Pilates.  (I do Pilates and water walking on alternate days, generally doing each twice a week).  This morning, I started to do that, then I thought about mindfulness, and hence, my dilemma. Should I pull in the abdominal muscles to strengthen them, or should I relax them to allow my lungs to breathe more fully?  Both have their benefits.  I was watching my kitten this afternoon (as I often do, because she is so cute), and noticed that her belly rose and fell when she breathes.  This appears to be the more natural way of breathing.  
Here is what I have decided.  When I am exercising, doing both the Pilates and the water walking, I will pull my belly in, because one of the goals while exercising is to strengthen my muscles.  At other times, I will let my belly relax.  This way, I will get the benefits of both