From a pure safety standpoint is there a functional difference between a climbing helmet, cycling helmet, etc? Is a cycling helmet actually the safest solution for a cyclist, or would the greater coverage of a climbing helmet be better for safety?
I would argue that there is a wide functional gap in safety helmets for various sports. The design tradeoffs are easily summarized:
Each of these design criteria have very inherent safety tradeoffs. In general, the type of cycling helmet that has a cosmetic and purely "hold the styrofoam together" type shell would not be appropriate for climbing where you might expect to have tens of rocks fall on your head during one climb. Similarly, the padding on a cycling helmet might be more than is needed for climbing. Looking at the spectrum from a full face racing motorcycle helmet to a winter skiing helmet, to a rock climbing helmet, to a skateboarding helmet, to a mountain biking helmet and finally to a typical road cycling helmet - you can see a progression to less and less armor as well as various amounts of styrofoam like insulation depending on the speed at which your body might hit the ground or an object if you fall from your vehicle.
I don't know how to precisely answer which is best, but hopefully this discussion helps frame your decision when picking a helmet if you can't get help from a subject matter expert that has experience in the activity you will use your helmet.
Safety equipment for each sport is designed, tested and certified specifically for that sport. Different sports have different and often competing priorities for the type of protection and comfort required.
An example of competing priorities might be that climbing helmets need to provide extra protection at the top of the head to defend against falling rock, whereas, cycling helmets are designed to provide additional ventilation for comfort.
In short, climbing helmets are not suitable for cycling and cycling helmets are not suitable for climbing. Wear the correct safety equipment for the activity you are taking part in.