I can't speak for usage outside North America, but as a motorcyclist in the U.S., I would say biking to mean moyotcycling is rare, and to bike meaning ride a motorcycle is even rarer.
Overall, motorcycling is not common in the U.S. It is very inexpensive to purchase, garage, and operate a car or truck compared to much of the world, so motorcycling has been relegated to recreational use, by a very small percentage of the population in most of the country. In contrast, bicycling has been a popular means of transportation for children for many decades, and is increasingly popular among adults. As such, to bike, to have a bike, to wear a bike helmet, to go biking to a bike shop, and so on will only be taken to refer to motorcycling where that context is firmly established, or if part of a compound which removes the ambiguity, like dirtbiking or a sportbike. Otherwise, most people will assume you are referring to human-powered bicycling.
One should also be careful with the word biker. While according to the dictionary this term could be used for any motorcyclist or bicyclist, in the U.S. it is strongly associated with the outlaw biker subculture and its innumerable hordes of poseurs. Motorcyclist or motorcycle rider would be the neutral alternatives. I am a rider, not a biker.
To ride is the most common verb for operating a motorcycle, but again, the context must be established that you are not riding a bicycle, scooter, or horse. Your second example is fine if you have established that "John" is a motorcyclist; otherwise, it would be safer to specify the mode:
Without thinking twice, John hopped on his Harley and rode to town in search of his suspect.
