There may well be differences between American and British English, or between sub-regions in either.
Where I live -- Michigan, USA -- we would typically say, "I ride a bicycle to work." If you said "I cycle to work", people would know what you meant, but it's not something we say here.
"Bike" can refer to a bicycle or a motorcycle, but, at least among people I talk to, it more commonly means bicycle. If you said, "I bike to work", I'd understand you to mean that you ride a bicycle.
But it could mean "motorcycle" in the right context. Like if we were talking about motorcycles, I'd assume you meant a motorcycle. Like if you said, "My bike is a Harley", I'd take you to mean motorcycle because I've never heard of a Harley bicycle. (Is there such a thing? I suppose it's possible.)