Dynamic range refers to the difference between the softest and loudest elements of the signal, expressed as a ratio. So yes, it is a reference to amplitude.
Many modern recordings, especially in pop/hip-hop/dance genres, are highly compressed (that is, they have a reduced dynamic range). In my view, shared by many, the sound quality of such recordings is dismal. But, as you touched upon, it is certainly true that in a high-noise environment, the reduced dynamics make it easier to hear both ends of the spectrum. That's why Muzak recordings were so highly compressed. You might find a search on "loudness wars" interesting.
A related measurement is crest factor...the difference between the peak amplitude and continuous/average amplitude of a signal. Good stereo recordings will have a crest factor of at least 10-12dB, or more (sometimes quite a bit more, especially in the case of classical music). Personally, I find good dynamics to be critical to sound quality...but unlike some others in this thread, have not concluded that high-sensitivity speakers are the only way (or even best way) to achieve them...especially when also considering other variables in the sound-quality "equation".
In any case, however, it always starts with the quality of the recording. A recording incorporating heavy-handed dynamic range compression will never sound as good as one approximating the range of a live performance. The next step is getting that recording, in the form of a complex electrical signal, amplified in a minimally unadulterated form, to the speakers. At that point, speaker quality takes over. And of course, at each step in the chain are the subjective likes, dislikes, preferences, etc as regards the constitution of "good" or "accurate" sound. For example, some folks may prefer a highly detailed, to the point of "hyper-realistic", sound quality. Others may prefer a rougher, "live venue" kind of sound. Etc etc etc.