Ikeda Sensei shows a number of basic techniques using "ki" and/or "aiki" and/or "kokyu-ryoku" (they're all elements of the same basic thing) ... some people are using the term "aiki", but "aiki" is just a usage of kokyu-power.
I agree with Jon that there needs to be some clearer definitions and I think (IMO) the place to start is with "kokyu". ... If there is going to be a discussion on defining the facets of Aikido, IMO the place to start is with the term "kokyu".
Watching and feeling Ikeda got me started on my own way of looking at these things, and agree or disagree with me, that's where it started.
Leaving aside my precise mechanical conclusions, let me observe something about why "breath" and "kokyu" (in the operative sense that we use it, which often looks like it has nothing to do with breathing as such), are properly descriptive.
We inhale and we exhale. Neither of these processes is passive. The sensation of breath is of two forms of "pressure" in the structure of the body. While these "pressures" are inverse to one another (obviously), nevertheless the sensation of "pressure" in the body is felt to be similar in both cases even though the action is inverted.
I can wax mechanical about the why and wherefores, but that basic sensation of pressure of this type which we operatively call kokyu, is captured in the mechanism and actual sensation of breathing. The extension of this sensation to the action of the remainder of the torso and limbs, and ultimately into a connected partner is a key aspect of the training.
When some traditional sources talk about breathing in on one side and out on the other they are speaking of a superposition of these two pressures in different lines of action in the body that actual breathing does not itself accomplish, but the description, which is otherwise nonsensical , correctly and concretely identifies the sensations sought to be achieved.
My effort hes been to define the nature of the inverse "pressures" and their interaction in the body, but I will leave it at that.