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?