Having trouble with implementing circular player movement

Hello all. I'm pretty new to Unity and a complete beginner when it comes to programming, so I'm looking for some help with a problem I'm having.

I'm trying to create a control method whereby the player moves in a circular motion, along the path of a several concentric rings. I was able to get movement from ring to ring working, but I'm having trouble getting the circular motion working correctly.

After a couple of false starts I found a script here which seemed to be just what I needed. It works for the most part; however the problem I'm having is that any left/right movement will gradually cause the player to drift out of orbit.

Here's a web player link, it may describe the problem better than my summary.

I also tried an implementation whereby the player gameobject is parented by an object in the centre which does all of the rotation, causing the player to rotate around it; but this gives me trouble when changing the player's distance from the centre. The speed increases the farther out the player moves, which is undesirable, as I need the player to have a uniform movement speed. Web player example of this attempt.

Does anyone have a solution which could help?

Edit: Fixed links.

Thanks for reading,
Ben Wilson (zerofiftyone)