If you add DSP to the audio after the demodulator, then that seems to be a radio with an audio processor, not an SDR, as nothing is done in software in the radio/RF signal path.
But if a radio adds digital sampling and DSP to even at a very low IF (VLF or ELF) and before the listenable audio demodulation, it's hard to tell that architecturally or signal-path-wise from newer or current (much higher IF) superhet SDRs, except by the lower digital processing and ADC sample rate requirements, as limited by the earlier technology.
So it's a matter of how you slice the definition between audio processing (audio CW filters and TNCs, etc.) and SDRs.