The article is very poorly written, but here's an explanation of what they're saying. An "inductive Turing machine" is a Turing machine which is allowed to run forever, but for each cell of the output tape there eventually comes a time after which it never modifies that cell again. We consider the machine's output to be the sequence of eventual limiting values of the cells. Such a machine is strictly more powerful than Turing machines in that it can compute more functions than just recursive ones. In fact it's an easy exercise to show that a function is computable by such a machine iff it is "limit computable", meaning it is the pointwise limit of a sequence of recursive functions. Limit computable functions have been well studied in mainstream computer science, whereas "inductive Turing machines" seem to mostly be used by people who want to have weird pointless arguments about the Church-Turing thesis.