It's my understanding that DSP is any signal processing that occurs in the digital domain.
On the other hand, some manufacturers tend to use DSP as though it's a proprietary term for whatever process they may be using, so for a lot of end users, it connotes what their equipment does in terms of DSP.
As an example, I have several pieces of Meridian gear. One is a 518 which is a digitial-to-digital processor for upsampling, volume control, filtering, etc. Another is my 565 is decoding formats (like DD, DTS, etc.), has embedded soundfield options (Trifield, Chamber, etc.), controls volume, tone, delay, polarity and some other odds and ends. The digital speaker can also perform some processing such as bass, treble, tilt, room boundary compensation, etc.) IMHO, they are all DSPs, but with different functionality.
Clear as mud?