Turing's definition of computable reals can't be wrong, because listing decimal digits is equivalent to giving rational approximations. The problem here is a confusion between real numbers and Turing machines that compute them.
This has nothing to do with such a "confusion", by the way, which is obvious to anyone writing about this stuff (including me). Rather it's the difference between something being logically definable and computable, which Turing himself writes about in his paper.
Wrong is a judgement call, of course, but given that we're talking about computability here I think it's fair to call Turing's definition flawed. There's a reason nobody uses it in computable analysis.
[1] H. G. Rice, Recursive real numbers, Proc. Amer. Math. Soc. 5 (1954) https://doi.org/10.1090/S0002-9939-1954-0063328-5
However, under Turing's definition there is no algorithm that can uniformly compute addition or multiplication! This has nothing to do with whether the set of computable reals is closed under those operations (although it is, of course - this is an easy exercise, no need to cite a paper).
I'll be more formal. Let us say a Turing machine A is a "Turing-real for x" if A outputs the successive digits of x when run. Then consider the following problem: given two Turing-reals A for a and B for b as input, output a Turing-real for a+b.
It turns out this problem is incomputable!
On the other hand the same problem but for the modern "approximation-reals" is computable, a marked improvement.
The lack of computable addition and multiplication is the flaw in Turing's definition. This is well known stuff - there's even a discussion of it on the Wikipedia page for computable reals. The fact that you keep bringing up orthogonal issues, like closure of the computable reals and the distinction between a Turing machine coding for a real and the real itself, makes me think that you might be missing the crux of the matter here.
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.