Enumerate over all halting Turing Machines?
The paper you're referring to is using "enumeration" just to mean "ordered list". Your feeling that such a list can't be computed by a Turing machine is correct. However, the list exists, even though there's no algorithm that will tell us what its members are.

Also, note that it refers to "an enumeration of halting Turing machines", not "an enumeration of the halting Turing machines". So it's just any ordered list of halting Turing machines, not necessarily all of them. It specifically mentions enumerations of polynomial time Turing machines. You might be surprised that these are enumerable, up to equivalence*. This is because you can turn any Turing machine into a polynomial time Turing machine by picking your favourite polynomial and modifying the machine so it rejects if it hasn't in that many steps. Since you can enumerate all polynomials (say, with nonnegative integer coefficients), and you can enumerate all Turing machines, you can combine the two to enumerate all polynomial time Turing machines.

* Two Turing machines are equivalent if they give the same output for every input.