I think the definition is going to depend on the location, what city, on public or private property, etc.

In my former neighborhood, a car left on the street without appropriate parking zone stickers could be towed after a couple of days (usually they were lazy and just booted the wheel).

In another neighborhood, a car with even the appropriate stickers left in derelict condition (flat tires) was slapped with a warning notice and towed when there was no response and he car remained unmoved (neighbors complained.) With impoundment, the clock starts on the time allowed before lawful sale/scrap of the vehicle.

On a university campus, if private property, they seem to be able to set rules as they wish. Bikes left after close of term could be considered abandoned, regardless of condition. They might require a property tag on bikes to give notice of suspected abandoned property. Obviously bikes deemed improperly secured, on railings, in access areas or in hallways or near building exits could be cut and removed without notice. But on a rack? Usually there would have to be some indication of abandonment, as in parts missing or general disrepair to indicate the bike was not rideable. I think that is generally up to discretion of people like campus security officials and/or groundskeeping and maintenance management personnel.