It's not quite as effective since you have to hold the dumbbell with your feet. Calf and toe raises will work the shins a bit as well, but the move described above, with dumbbells, hurts pretty fast. Like many exercises, it all comes down to proper execution of the movement. Only bring your knee joint to 90 degrees while lowering the weight and don't lock out the knee when extending. The vertical leg press is also more likely to cause back stress than the incline variety.
Donkey raises work the same way as standing raises since the knee is straight.
The reason for the different muscle feeling the stress is related to the angle of the knee and it's relationship to the origin and insertion of the muscle. To test this do a calf stretch. The straighter the knee joint the higher you feel the stretch. As you bend your knee you feel the stretch lower in the calf and down into the achilles tendon. Sitting down in a chair with your feet flat on the floor raise your heel and flex your calf.