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There's a zebra crossing on my commute, which forms part of the shared path from Bristol Temple Meads to the centre. Cars generally stop there for cyclists. Further along the shared path, just before you hit Queens Square is another crossing, this time no zebra crossing but a raised section of road (essentially an extended speed hump) and 'give way' markings on either side of the hump for cars. Its strange because it is a much more ambiguous crossing for everyone, cars, bikes and peds. Often cars will give way, but about 20% of the time they don't. I've often wondered why they went with that there and not a zebra.
Cyclists are allowed to use zebra crossings but the point is whether (AIUI) cars are required to stop for cyclists who are riding (or even sitting upon) their bikes and want to use the crossing. I don't believe they are.
Car drivers would not be aware of whether there is a cycle/shared-use path either side of the crossing so I think that is a red herring.
The Crank vs Brooks court case shows that someone wishing to use the crossing and just happens to be pushing a bike should be treated as a pedestrian. (But such a ruling does not tell us anything about people who are riding their bikes at the time.)
The only certainty is that riding along and attempting to swing onto a zebra crossing and expecting all motor vehicles to stop and give way is a very risky strategy.