A drum synthesiser is a synthesiser designed specifically to make drum sounds.
Pre-MIDI analogue examples were often designed to be triggered via one or more drums, either as a whole drum kit in their own right, or that could be added to an existing acoustic drum kit. Although not designed for the purpose, they could instead be triggered from a digital step sequencer.
MIDI-era digital examples were designed to be more likely triggered from a MIDI sequencer, though they could still be triggered by a drummer using MIDI controller drumpads.
Whether they were designed to be played live or sequenced says less about the type of interface used, and more about the fashion of the music during the era of that interface. Microprocessors had enabled both MIDI to exist as a concept, and complex sequenced music. That MIDI-era equipment expects quantised playing merely shows that the state of the art was advancing in multiple areas at once, due to the same underlying technology.
Drum synthesisers are often conflated with drum machines, although I'd argue that the category of drum machines would be more useful if narrowed to only include sets of drum synthesisers that are paired with their own internal step sequencer, such as the TR-808.
- Pearl Syncussion-1
- Simmons SDS-V