As a non-professional cyclist, I'd assume 'off-road bike' to be the motorised version. If they meant a cycle, they'd say 'mountain bike.' I doubt, for once, that I'm unusual in this assumption.
As a cyclist I'd use "mountain bike" for a pedal bicycle designed for off-road riding (unless I was being particularly technical), and I'd only ever talk about pushbikes in the context of the Birmingham cycling campaign group, but I wouldn't assume a non-cyclist would get the terminology correct.
Similarly, if you say 'scooter' I'd likely think of a skateboard with steering, of the type popular with teenagers of all ages. But a motorcyclist or police officer might think of a class of motorcycle.
I've also learned to say 'bicycle' rather than 'bike' around QUILTBAG women, lest they apply the wrong set of stereotypes.
It's all about context. And news headlines are notoriously (and often deliberately) bad at that.