Carcassonne is also really nice with children. You can start them on just the "puzzle" aspect on attaching matching tiles, without scoring.
Our oldest child is now capable of the base game, and I can still make it interesting for me by going for secondary objectives, such as filling difficult gaps ;-)
In particular, the core disagreement seems to be about whether the automorphisms of C should keep R (as a subset) fixed, or not.
The easy solution here would be to just have two different names: (general) automorphisms (of which there might be many) and automorphisms-that-keep-R-fixed (of which there are just the two mentioned.
If you make this distinction, then the approach of construction of C should not matter, as they are all equivalent?