The subtle mind is all about breathing. In the subtle mind if you still the breath you still the mind. A peaceful breath is a peaceful mind. The first concentration is to use your breath as your focal point, although, I find it so hard to keep a focus on the breath. My mind runs all over the universe. I tried to settle into the natural ease of my body, returning my mind back to the breath or my anchor/focal point, when my mind or thoughts would wander, but it is hard to keep taming my busy mind. It is a process like the CD said, and it will be something I must get used to. Being patient and not so much reacting to everything. I am supposed to witness things without reacting to them. In our witnessing consciousness or healthy psychology, we can stop, see, understand, make choices about our thoughts and feelings and have clearer intentions (but no quick reactivity anymore).
We can see it/things/thoughts/feelings/images, whats in our head rolling around, our mental activity, but not grasp it, just see it. We can still our mind. Not hold on to the mental activity, just see it and let go of it and let it fade away and just witness it come and go. We can control the mental chatter, shut it down, turn it off and be in stillness again focusing on breath only.
This is how I understand the Subtle mind. It is really a control of my emotions not grabbing a thought, especially a disturbing thought, and running with it. I can see it, acknowledge that disturbing thought or image, and let it go. It is like I am in control. My thoughts do not control me anymore, I control them. I don't react.

If I were to compare the Loving-kindness(LK) practice, with the subtle mind exercise, I like the first one more. In the loving-kindness practice, it is enjoyable to think of a loved one, and open my heart to their thots. I do not like to think about the bad sensations, as it suggests, but, I try not to grasp at it, as the CD says. This will take me more time to learn for me. It is easier to give the good thots and images more attention. Letting a loved ones suffering come into my heart and body is hard for me. I would rather think it than breath it. I would rather think of help and joy for them than breath it in and out of my heart and body. This is a whole new concept, not easy or sure I can do this. Imagining a circle of strangers and taking in their suffering is hard. I can imagine someone I know better. I can do it if I imagine a flood, or landslide and get sad for the people on TV that this happened to. (I do cry when I see things like that on TV, does that count I wonder?)
Observing and not grasping. Loving myself, feeling a natural rest, ease, and openness is hard for me, but necessary according to the LK practice. I have thot of others more in my life and turning it to myself is harder for me.

Both practices do not allow grasping the thots, feelings and/or images of the mind, but, just observation. The natural home is awareness of and not the grasping of what we observe with our mind.

I take the mind-body connection to mean that we need to keep our mind stable, not upset or stressed, but under complete control. Our body and the way we treat it, needs to be controlled. If we don't control what we eat, we will end up weighing 400 pounds. If we do not eat healthy, we could get cancer or die of a heart attack, or high blood pressure could lead to a heart attack. Our exercise, diet, habits, all need to be balanced with our mind and emotions to have a flourishing life. I personally ride a bike many miles a day (26 today) to keep a mental and physical balance. I feel successful and productive after I have done my routine. This makes me feel balanced and keeps my body balanced.