Is there a term used?
How about aiki.
I have also felt this intention from my partner, and like you, it first became noticeable in weapons training, but through the years it always seemed hit or miss; as if I had no "control" over it. I began heavily focusing on it about 8 months ago after attending a James Williams seminar, and have become much more sensitive to it. Subsequent insights I had totally redefined ki no nagare practice for me.
Really it made me laugh, because I had always thought of myself as a fast learner, but it struck me that I only learned the easy stuff fast! That's ok...
If you think about it, it is quite obvious that the intent to act must precede any act. And that this intent is much different from just thinking about acting. It is what actually makes the body move. If you further realize that it is not a one time occurance but a continuing event, you can watch it become commitment (or not).
I also currently feel it is important to define ki as spirit... the life in the body.
I see aiki as a principle of relating or interacting, which allows one to still follow the opponent, so as to be appropriate, and yet does not require waiting for the physical act before responding, so as to avoid being late.
If you place yourself in the context of a sword fight, where the first contact will almost surely decide the outcome, What are your other options for a successful interaction? Luck? Overwhelming speed?
I can't speak on the ideas in book you mentioned, but you'd probably be best off just intensely focusing on this yourself.
You might also find this video inspirational
. Feel free to watch the whole thing, but most pertinent to this topic is from 16:15 on. It is not aikido, but Anderson Silva is doing some interesting things.
I'm no master, but hopefully this helps a bit. I'd be happy to discuss this further.
Great topic, great post!
Let us know what else you discover.