In the field of computability theory, the terms "decidable", decidable set", "computable", computable set", and "recursive" recursive set" are all formally defined and they all mean the same thing. So, to put it gently, Wells is misusing terminology.
There is an enormous amount of evidence that the formally-defined class of computable functions on the natural numbers is exactly the same as the informally-defined class of effectively calculable functions: the ones for which there is an effective procedure that could, in principle, be carried out by a human with unlimited time, resources, and patience. That is, there is an enormous amount of evidence that the Church-Turing thesis is true in the form "a function is effectively calculable (in the informal sense) if and only if it is computable (in the formal sense)."
Regarding Wells' invocation of Tarski, we'll never really know what Tarski had in mind, because Tarski died within a year of the conversation Wells describes. But Wells' argument that an entire field should redefine the formal term "decidable" based on an off-the-cuff discussion with Tarski is not compelling.
There is a research area known as "hypercomputation" that studies various models of computation that go beyond things computable by Turing machines. The reason that this work is not viewed as evidence against the Church-Turing thesis (as stated above) is that these models don't possess the essential property of being able to be carried out by a patient human using unlimited paper, pencil, and time.
That kind of effectiveness is the heart of Turing computability, and the reason that we use the word "computable" to refer to the Turing computable functions rather than some broader or narrower class of functions.