Believe it or not, both technique books are trying to describe the same exact technique -- In fact, there is a third method, TH, BH -- but none of them is a totally accurate description. The problem is the method of documenting the technique, and this is a perfect illustration of one of its shortcomings.
Bronze Foxtrot rise & fall is unique in that the lowering is much earlier than most other side-close actions in the swinging dances. Whereas in Waltz, the rise continues until the feet close, the rise in Bronze Foxtrot hits its peak when the weight is split between the feet, and lowering is complete by the time the feet have closed. If one were to draw the shape of the rise & fall over a series of continuous chasses, it would appear as a series of arcs (like very shallow McDonlads arches), where the bottom points of each arc represent the point of foot closure.
Everybody has a different opinion about how the feet should be described, because it's difficult to be accurate with such an oversimplified description model. The truth is, both heels lower and then touch the floor at exactly the same time, but the description is only capable of addressing the action of one foot at a time. So what you're reading is various differing opinions about what description seems to come the closest. (My personal favorite is TH-BH, which was the Fred Astarie team's attempt to show that the feet are partially lowered by the time the first foot reaches the second).
At any rate, the true answer is (D) None of the Above. But if you absolutely *must* use the generally accepted methods of documentation, then the correct answer would be the one that makes your examiner happy. If you're not taking an exam, then rather than thinking of one foot at a time, just remember that the heels lower together, at the same time and speed. You are at the peak of rise at mid-stride, and fully lowered when they're closed. That's it!
Regards,
Jonathan Atkinson
www.ballroomdancers.com