I'm no expert but I think concentration is only part of the picture. This can be concentration on the breath, or concentration on a physical activity as you describe it. The other part is using the highly developed concentration state for the actual (mindfulness) meditation, which is pretty hard if you're moving around, at least for a beginner.
I'm talking about Vipassana practice, which is all I know about. As you say there are many forms of meditation.
edit: I used the word 'deliberate'. I should say seemingly deliberate. To be exact, it's not about deliberation or thinking at all.