In Switzerland we say Velo (deriving from French vélo, short for vélocipède ‚fast-foot.‘) to all sorts of bicycles. But of course cool bicycles nowadays are called "bikes", as for many things we use anglicisms. So there are mountain bikes, road bikes, dirt bikes, track bikes, touring bikes, e-bikes... But there is an unwritten rule: never, ever call a velo (bicycle) with a basket, a "bike".
Interesting to hear about your unwritten rule for velo bicycles in Switzerland.