The four strokes of a four stroke cylinder are intake, compression, combustion, and exhaust. That's down, up, down, up. Since there's no down or up in any of this engine's combustion chambers, "strokes" is probably inappropriate.
As you said, 4 stroke has intake, compression, combustion, and exhaust. That's expansion, compression, expansion, compression. It fires between compression and combustion.
A 2 stroke motor has compression and expansion. It fires between compression and expansion. Intake, exhaust, and combustion happen during the expansion phase.
A rotary motor is traditionally 4 stroke, and I'm not sure a 2 stroke version is feasible. The LiquidPiston rotary appears to be a 4 stroke to me, despite the article text.