"Something to do with the fact that planets don't shine by their own light but by reflected sunlight."

Fired up by my own curiosity I just checked this. I'm wrong. It's to do with the fact that the huge distances of stars mean that even the brightest star is seen as a single point of light even by the most powerful telescopes. On the other hand, planets are actually tiny discs, even in binoculars, and act in effect like several points of (less intense) light. The effects of refraction and turbulence of the atmosphere severely affects stars, being just single points (like having just one pixel excited), so they sort of go on/off in a twinkly way. Planets on the other hand, being bigger in sky area, are more like having several pixels which will not all be affected in the same way at the same time, so any tendency to twinkle is, er, averaged out. Nothing to it.