Algodoo

This has always baffled me. My logical brain wants to think: "The number of teeth on both pinion gears is equal to each other, and the number of teeth on the internal and external gears is equal, therefore the number of rotations that the pinion gears make in both gear sets should be equal to each other". But, it doesn't work that way!

I was thinking the same thing when I made the scene. Here's one way of looking at it:
Imagine you start with a straight rack and a 12‑tooth pinion gear.
Make the rack exactly twice the circumference of the pinion. If you roll the pinion along the rack from left to right, the pinion will make two full rotations by the time it reaches the end.

At that final position, freeze the pinion relative to the rack—as if you welded them together.
Now hold the left end of the rack fixed, and bend the rack into a circle so its ends meet. You can bend it so the teeth face outward (forming an external gear) or inward (forming an internal gear).

When the rack is bent into a circle, the gear that was “frozen” at the end of the rack is forced to undergo one additional rotation (for an external gear) or one fewer rotation (for an internal gear) in order to close the loop smoothly.

AI offers this simpler explanation:
Think of a planet gear rolling around a sun gear.

When a gear rolls around the outside of another gear (external mesh), it not only turns because of the tooth engagement — it also turns once more because it is physically walking around a circle.

When a gear rolls inside another gear (internal mesh), the walking direction is reversed, so it effectively turns one less time.

I like your explanation over the AI explanation. It makes sense and now my logical brain is satisfied!