Giles wrote:But if the push is based on a sinking of the body and a relaxing of the arms, then in my experience the transfer to short power is a lot easier.
One can generate force - "long" or "short" - by expansion/opening or by contraction/closing. In both of these there is sinking of the body, but at different points in the sequence. The arms should always be "relaxed", but not limp, regardless of opening or closing. So, your description doesn't, for me, isn't what distinguishes long from short power.
In some styles, the entire body doesn't do one thing. That is, parts of the body do one thing, open or close, while parts of the body do another, open or close, but in a coordinated way. In that case, some parts open, some close, some expand, some contract. That can be done while shifting weight or not.
If A can then project B backwards with a clear push and maintaining relaxed arms without A himself falling (or swaying) forwards or backwards in the process, then I find this translates quite well into short power.!
The salient point of the exercise you describe seems to be that there is no linear translation of the centreline, per Bao's point. There are probably lots of ways one could describe the distinction between long and short power. Bao's description captures some of that with his stating that long power involves follow-through and a translation (weight shift between legs) of the centreline, while short power doesn't. That distinction works for styles that do shift weight while pushing, but doesn't include those that don't (e.g. Hong style). In other words, its possible to push (long power) without necessarily shifting weight.
As you pointed out, duration of force transfer is one of the distinctions between long and short power. Another is that in long power, the object is to move the opponent through a distance; in short power the object is not to move the opponent.