Intent (mind or mindfulness, the "spark" of will) drives Ki (energy inherent in neuromuscular-biomechanical "internal" process), which drives the physical action (physical force or power). The coordination of them produces waza (outward expression).
This jives with my understanding, basically the intent is the mental control mechanism for the ki. To connect it to the OP discussion of breath and ki:
The intent is how you can affect the ki volitionally. But, the breath also affects the ki-- the pressurization can load/strain the ki. So, if you are aware that both of these (breath and intent) have effects on the ki, then you can play with and train the ki using the two of them. Then, in application, you have the tools to induce a "filled"* state and to manage it using the intent.
*couldn't find the original source but google led me to this blog post talking about the passage I was looking for:
O-sensei when he was sick was still practicing. The way his body acted when in the "on" state made him hard to lift, but he could turn it off at will. He just changes his intent, and that controls his "ki-filling" state.