Does the MT5600SMI-xx modules support distinctive ring?

Yes, Distinctive Ring is a voice mode feature that just reports the length of the ring pulse and the length of period of silence between rings. Output from the modem will look something like this when the feature is enabled:


The X and . are headers to mark the beginning and end of the ring event, in this case the ring pulse lasted 2 (DRON) seconds and was followed by 4.1 (DROF) seconds of silence.