DSP describes the mechanics of making sound through computation. (It literally stands for digital signal processing.) This means everything you described after it all fall under it. (Assuming the digital circuit you reference has a processor of some type in it, otherwise I don’t see why it would be a digital circuit instead of just a circuit in general.)
It isn’t a flavor, it is the main flavor all others are derived from. Just like how Ford is the manufacturer and they make escort, mustangs, Taurus, etc. car models. Those cars are all very different but they will all get you from point a to b.
That said, there are many different takes on DSP in action, but the processor used in those endeavors is not really the main point of distinction. The DSP code (methods) involved are first and foremost, and then the hardware matters, as it is was enables that code, and also limits it.
Lastly the whole is it analog or digital question is still an important question, but it is one that is becoming more and more irrelevant thanks to advancements in both hardware and DSP theory/code.