Motor or engine is silnik, although some people sometimes use the word motor to refer to an engine/a motor as well.
But by default and in most contexts, motor means motorcycle.
I've actually learned this word as "motocykl". Whereas "motor" seems to have a more colloquial flavor, I might be totally wrong though, as always.
I'd say that it should. "Bike" in English can be either a motorcycle, or a bicycle. "Motorbike" is also regular English; even "motor bicycle" is occasionally heard.