What is the difference between a balance exercise and a stability exercise? A balance exercise is training your muscles to achieve equilibrium, challenging different planes of motion. A stability exercise is exactly that, teaching your body to stabilize. Once your body stabilizes, it then learns to use stabilization to move your body as a whole properly.
Side planks fall under the category of stabilization, NOT balance. Most people execute a side plank by stacking their feet on top of each other and spending a lot of energy balancing (with their shoulder and ankle muscles, not their core!) and when you watch someone using this technique, most of the time it looks like they are practicing their wobble moves rather than holding a perfect plank. But a side plank is meant to stabilize, and use the muscles throughout your core—like your glutes, hips and back.
Next time you do a side plank, try this. Instead of stacking one foot on top of the other, place your top foot on the ground right in front of your bottom foot. Now instead of concentrating on trying not to tip over, you are working on engaging your core to keep your hips up and your body straight rather than trying to balance forwards and backwards.