What came to mind intuitively is what I would call C-T tests that are more or less methods of accepting some model as being a computational model or not. The question is in what amount and how could we balance the viability of the tests with problems of how effective they would be(both computationally and effective as being acceptable as testing methods) and depending on the answer would these raise circularity issues?
Computability theory overflows with hierarchies of computability notions, which allow us precisely to compare the computational strength of diverse notions. We have the hierarchy of Turing degrees, which measures the strength of relative computability by oracles. We have the hierarchy of complexity theory and the complexity zoo, which measures the strength of diverse resource-limited computation. We have the hierarchies of the arithmetic hierarchy and that of descriptive set theory, which can measure the strength of infinitary notions of computation, and the hierarchy of the constructibility degrees, which can measure the strength of stronger notions.
All these hierarchies are quite commonly used to gauge the strength of a generalized notion of computability, whether the notion is above or below the usual Turing notion. One gains an understanding of the strength of a given computational model by fitting it into the known hierarchies and thereby also coming to know how various models of computability relate to one another.