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I don't usually read or post in threads about embouchure, but, hey, I've only got one life, so let's do it.

I think of it as a simultaneous "engage" and "drop" action for playing in all of the registers, not just low. That is probably because I have an issue with just keeping my oral cavity open. I get the best results when I "feel" like the mouthpiece is as flat as possible on my lips, and I do that through an engaging of my lower jaw against the rim. That may also be because of my teeth; I don't know.

Getting an engagement to your jaw, though, is not something I would recommend, as there may or may not be legal, moral, physiological and grammatical implications therein.

I would agree that the term drop is overly ambiguous. I don't like the idea behind the word engage, as it sounds positive in meaning, and more rewarding psychologically to practitioners. While trying to corrupt/teach young tuba players, I usually spent more time just trying to get them to open their teeth with the phrase "more space between your teeth" being used. It also seems like the "drop your jaw" line gets players to pull the lower jaw back, which is wildly counter productive. "Open your teeth" seemed to recall experiences with the dentist, and resulted in keeping the lower jaw forward. "Engage" would also seem to keep a better balance of contact between the upper and lower jaw when compared to the "drop your jaw" concept while not recounting visits to the dentist (although some can be mildly pleasant with a well-formed hygienist).

Taking it a step further:

This goes along the same lines of how the lower register is often mis-handled with younger players. Just like F tuba problems, a lot of them neglect a firm structure for their embouchure in all registers. In the low register the tuba can be relied on to slot (somewhat) the pitch and players rely on the harmonic spacing rather than precise buzzing to control pitch. When they develop a strong and centered buzz, from a well-developed and *firm* embouchure, the low register starts to really come alive. If you look at any of the guys out there that are known for extreme low register strength, you will see firm corners on notes well into the pedal register. No "let loose and hope for the best" technique really provides enough focused core to the sound.

I think of it as overbite for upper range, underbite for lower range. Engage seems, to me, to be better for the lower range, seeming to mean get it more into the act.

Good stuff!