Not quite. Adaptive V-sync turns v-sync off when you drop below your refresh rate. Regular V-sync, when using a double buffer system, is the one which often drops to half FPS, not because it's a rule of regular v-sync, but just what happens as a result of not letting the GPU to work on a new frame until the finished one is displayed.
Anyway, V-sync and adaptive V-sync stops tearing and locks your FPS to no more than your refresh rate.
Adaptive V-sync turns V-sync off if you fail to reach your refresh rate, so tearing will occur at that point.
V-sync will not allow tearing regardless of FPS, but can result in much lower FPS, often half FPS if you are not able to reach your refresh rate in FPS (this happens with double buffering, triple buffering fixes this, but can result in stutter).
Adaptive V-sync has an option to lock FPS to half your refresh rate.