Stan,
Do you want me to explain it or justify it? Because frankly, I can't justify it... it is a mess.
But then again, it's just reflective of different regions and communities developing variations of the same or similar dances. How else, for example, would you explain the many styles of Tango, which are so dramatically different?
There's actually a little bit of history behind the two variants of American style Rumba timing. Apparently, when Arthur Murray first introduced the concept of the Box Step and the "inter-related system of teaching", he was somehow able to acquire a patent. So as a workaround, the Fred Astaire school came up with the novel idea that if you start the box with the chasse and count it QQS, it's an entirely different step. We all know better, but I guess it flew in court.
As for the break on 2, I've heard a few different stories about how that came about. One states that it developed naturally from the Clave, which accents the 2 beat in the odd measure. Others I've heard say that dance teachers plotted to make it a required timing, simply to make it more difficult (and thus, require students to take more lessons). Frankly, I don't know if I buy either of these, but they seem to be the most prevalent explanations.
In the end, the timing you choose should be the one that you find to be the most musical. And that always depends on the piece of music that you are dancing to. You may also want to take into consideration the cooperation factor... If you hope to get along with the person you're dancing with, it's best to dance the timing that you think she's used to. The only time you're actually required to use any specific timing is during tests and competitions.
Sincerely,
Jonathan Atkinson