What makes a counterexample exemplary?

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Proof constructions and evaluations (2009, methodology): “…that were in accord with prospective teachers’ “example spaces for validation” at a given time (building on the work of Watson & Mason, 2005; Zazkis & Chernoff, 2008), and (2) incorporated activities in the instructional sequences that helped prospective teachers become more aware of…”
Paradoxes as a window to infinity (2008, background): “Paradoxes regarding infinity present learners with a potential cognitive conflict (Zazkis and Chernoff 2008) where one’s incompatible and inconsistent ideas are recognised by the instructor, but not yet recognised by the individual which can develop into a cognitive conflict in an instructional…”
Scaffolding Based on Cognitive Conflict in Correcting the Students’ Algebra Errors (2018, background): “Based on the research results, it is known that cognitive conflict plays an important role in changing the students’ concept by examining the mistakes made by students (Zazkis and Chernoff, 2008).”
Facilitating the Transition from Empirical Arguments to Proof. (2009, background): “5 Strictly speaking, Zazkis and Chernoff (2008) did not use the term pivotal counterexample.”