It's not obvious for me why and when bike developed as short form from bicycle. Could you explain that, please? And is it odd to say mountain bicycle or motor-bicycle?
This link would seem to be the definitive analysis of this contraction, from bicycle to bike (preview only without subscription):
There is, however, a possible explanation from generative phonology which is quite straightforward. Generative phonology postulates underlying segments that differ from their phonetic realization if doing so will allow generalizations that could not otherwise be captured in the grammar. ....
Apparently, contrary to popular belief, bicycle did not come from French to English, but rather most likely went from English to French.