I think I may have just enough insight to offer a small explanation of my own, thanks to the teaching of several excellent teachers (in aikido and other fields). You may not believe it, or perhaps you may find exceptions. But, on a small scale, this is what I and others have observed.
It is possible to send out energy/attention or ki that is not positive - ki or energy that is sent with anger or hatred. I do not think that one is relaxed, centered, or weight-underside at such times. In that sense, I think the phrase "If you are extending ki, you are automatically centered and relaxed" is not quite true. People routinely send out energy when they are tense and hostile. (Maybe the problem lies in my defintion of "ki"? That's quite possible.) On the other hand, if one is truly relaxed and truly centered, one can not be simultaneously angry or full of hate - one is rather calm and peaceful - and the energy one is extending at such times is usually "positive" ki. (Sometimes it is said that one's eyes should be "soft" and not "hard" when practicing aikido).
When I was in George Simcox's Virginia Ki Society classes, he did an occasional demonstration of some of the power of "positive" ki. I think Harry Eto Sensei also did a similar demo at the 1995 Maryland seminar. You may demonstrate these for yourself if you already know the basics of ki testing. Note: "ki testing" is nothing more than applying physical pressure with one's hand to someone (such as pushing gently on the upper back, shoulders, or near the collarbone), in a test of stability - however, there are many different ways to apply the force, some much much harder to withstand than others. It also must NOT be done with an attacking or critical mindset, because then it is no longer a test but an attack.
The DemonstrationsAs a note, these demonstrations have not been subjected to rigorous scientific testing. However, it would be interesting to do these tests at some point.
Demonstration of the Power of Positive Ki - Effects on the TargetYou'll need at least two people, preferably 3, for this. Preferably, all should have some ki training (even if it's just a 5-minute demo of how to do ki-tests).
- Stand one person in front of the other(s). Ki-test this person with level-one tests, gently pushing on her (with muscle strength, not with ki) from the front, back, and sides to establish her level of stability.
- Instruct the observer (or class) to look at this person (who is probably nervously smiling at this point) and to think how truly wonderful she is. A great person, with a friendly smile, someone good to know, and an excellent friend.
- As the observer(s) are doing this, ki-test the person being observed again. How does she fare? We've found that such subjects do much better when tested under the positive gaze of other people.
- (It is possible to do the reverse of this, of looking at the person and thinking negative thoughts at her. This seems to have the effect of making the person less stable. However, I don't recommend doing this very often.)
- (This experiment has been done with the person being looked at not knowing whether the people looking at her were looking at her with positive or negative thoughts. This was not rigorously controlled, however. In any case, the results were the same - negativity weakens, positivity strengthens.)