Consider the halting set; … is not enumerable / computable.

…

Here, we should be careful with how we interpret “information”. After all, coNP-complete problems are trivially Cook reducible to their NP-complete counterparts (e.g., query the oracle and then negate the output), but many believe that there isn’t a corresponding Karp reduction (where we do a polynomial amount of computation before querying the oracle and returning its answer). Since we aren’t considering complexity but instead whether it’s enumerable at all, complementation is fine.

You’re using the word “enumerable” in a nonstandard way here, which might indicate that you’ve missed something (and if not, then perhaps at least this will be useful for someone else reading this). “Enumerable” is not usually used as a synonym for computable. A set is computable if there is a program that determines whether or not its input is in the set. But a set is enumerable if there is a program that halts if its input is in the set, and does not halt otherwise. Every computable set is enumerable (since you can just use the output of the computation to decide whether or not to halt). But the halting set is an example of a set that is enumerable but not computable (it is enumerable because you can just run the program coded by your input, and halt if/when it halts). Enumerable sets are not closed under complementation; in fact, an enumerable set whose complement is enumerable is computable (because you can run the programs for the set and its complement in parallel on the same input; eventually one of them will halt, which will tell you whether or not the input is in the set).

The distinction between Cook and Karp reductions remains meaningful when “polynomial-time” is replaced by “Turing computable” in the definitions. Any set that an enumerable set is Turing-Karp reducible to is also enumerable, but an enumerable set is Turing-Cook reducible to its complement.

The reason “enumerable” is used for this concept is that a set is enumerable iff there is a program computing a sequence that enumerates every element of the set. Given a program that halts on exactly the elements of a given set, you can construct an enumeration of the set by running your program on every input in parallel, and adding an element to the end of your sequence whenever the program halts on that input. Conversely, given an enumeration of a set, you can construct a program that halts on elements of the set by going through the sequence and halting whenever you find your input.

Formalizing the intuitive notion of effective computability was exactly what Turing was trying to do when he introduced Turing machines, and Turing’s thesis claims that his attempt was successful. If you come up with a new formalization of effective computability and prove it equivalent to Turing computability, then in order to use this as a proof of Turing’s thesis, you would need to argue that your new formalization is correct. But such an argument would inevitably be informal, since it links a formal concept to an informal concept, and there already have been informal arguments for Turing’s thesis, so I don’t think there is anything really fundamental to be gained from this.