Proprioception is the CNS process of determining the relative position/movement of the limbs/trunk while balancing. Motor control, or neuromuscular function, determines the outcome, that is, a specific performance parameter. It is this aspect that most likely changes with training.
Performing proprioceptive exercise on a balance board is really a misnomer. Attempting to balance while standing on an unstable surface does not selectively train the proprioceptive system, it reweights the vestibular and visual information (versus somatosensory input) to a different degree than standing on a stable surface.
Vuillerme et al. demonstrated that gymnasts were able to reintegrate proprioceptive information (sensory reweighting) more rapidly than other athletes after tendon vibration at the ankle, suggesting that their specific training, particularly with regard to balance, could affect their ability to process proprioceptive information.
Ogard, W. (2011). Proprioception in sports medicine and athletic conditioning. Strength and Conditioning Journal, 33(3), 111-118.
Wacky insoles are a form of tactile proprioceptive stimulation.