My mistake. I see my oversight now. `Either String String` is not equivalent to `String | String`, but to `Left String | Right String`. The same must be done for the C# version.
Yes, you must have individual constructors for the left and right cases in order to distinguish them. In C# you would use two distinct record types for this. Haskell’s syntax is more concise though, since you define the constructors inline in the declaration of the sum type.