I think "controlled" is the key. It's got to have punch and impact. In other words quickness, or speed of attach and decay matters more to me than any other single factor. This all goes to the "control" issue, because when bass notes linger when they ought not linger, they get "muddy" or "flabby" and thus uncontrolled.
To tell you the truth, I don't think "deep" matters as much most people seem to think it does. In other words, if a pair of speakers or headphones can produce tight, controlled, accurate bass down to 30Hz, that's good enough for me. There is not a whole lot of information to be found beneath 30Hz, and hardly anything beneath 20Hz. Beneath 18Hz, I can't even hear it on test tones.
For example, on my Martin Logan Aerius speakers, I'm pretty sure that they don't go below 40Hz, and although they're most definitely bass shy without subwoofer support and a good crossover, they're quite good at what they do produce in the midbass region. When I've had them placed in my 2 channel system without a sub, they perform quite capably on 90% of all CD's.
We now have a host of new headphones claiming to produce bass down to 5Hz. Good golly, Miss Molly! This is what my Buttkicker (bass shaker) does! This is in the region of "tactile transducers", something to be felt, but not heard.