Difference between a 2-Cycle and 2-Stroke Engine

There is no difference between a 2-cycle and 2-stroke engine. The terms mean one and the same thing. By definition, a 2-stroke engine is an internal combustion engine which makes two strokes for each power cycle. The piston moves twice (up and down) to make the crankshaft rotate once.

It is commonly contrasted with the 4-stroke engine that makes four strokes for each power cycle. Instead of separate compression and combustion strokes like in the 4- stroke, the 2-stroke combines them in one stroke.

2 stroke engine
A 2 stroke diesel engine

You can look at this video for an explanation on how this type of engine works: