Okay, I see. I guess from my point of view, it's not a minor detail. It is simply an unworkable definition if you want to study computable reals in any meaningf

Okay, I see. I guess from my point of view, it's not a minor detail. It is simply an unworkable definition if you want to study computable reals in any meaningful way (i.e. beyond the absolute basics like defining the set of computable reals). By the way I read through that paper you linked and I think it's possible Rice wasn't aware of the issue here either. He references Turing's paper and notes that Turing's definition is equivalent to his, but what I've been getting at here is that this equivalence itself is not computable. There's no effective procedure to convert between Turing-reals and Rice-reals, even though you can prove that they define equivalent subsets of R. Turing himself proves this in "On Computable Numbers, with an Application to the Entscheidungsproblem: A Correction", so it seems he came to realise his error.