The strokes have nothing to do with up and down (e.g. flat motors, like a boxer), but to do with compression and expansion. The direction of compression doesn't really matter aside from packaging concerns.
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.
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.