This question is very usefully provocative, as evidenced by the comments, and the pro-example versus [sic] pro-abstraction notions... and the apt comment(s) suggesting that, in particular, the genuine issue is not that students have not "learned the definition", but that their intellectual methodologies are inadequate to the task... or maybe some really are a bit lazy or work-avoidant (sounds better!) while not being actively opposed to the mathematics.
E.g., I was a bit baffled by the comment that implied that it is perhaps dangerously deficient to characterize things as: continuity is that the graph has no jumps, and a differentiable functions' graph has no corners... considering that this is exactly my own intuition, although/and I can conform to the ambient standards-of-proof as need be.
I would claim that the underlying problem is that students are typically rewarded more for compliance and conformity than for other forms of astuteness. Thus, they have often learned to ignore their own intuition, since the instructor/exam/homework is actively trying to prank them in that regard. "Definitions" are therefore not at all codifications of extensive prior experience of serious people, but just set-ups for pratfalls of the victims, ... EDIT: for example, prank questions might have very little mathematical content, but, rather, play upon delicacies of wording, or the tension/conflict/ambiguity between mathematical use and colloquial use. No way to reason these out. Rather, one must know what the examiner is thinking. Similarly, textbooks can create conventions/rules/definitions with little genuine mathematical content, but, nevertheless, with very precise boundaries, the latter lending themselves to questioning.
In particular, in all my experience with standard courses/textbooks/exams, it is not the case that definitions [sic] are helpful, much less clarify pre-existing ambiguities or troubles or confusions. Instead, for exam and such purposes, they are far too often (and, then, this "taints the well") used to prank students in a quasi-legalistic sense. An example of the worst sort of crap is any question about "how many axioms does a group satisfy?". EDIT: again, such a question has essentially no mathematical content, because any finite list can be put together into a single axiom by conjunction, of course. But students may be simply required to remember what a particular book says on a particular page (as opposed to intrinsic mathematical assertions). Similarly, notational conventions can be "tested".
The other issue, which is maybe not literally "laziness", but "passivity", is (I claim) partly a result of years of being beaten down by petty-despot "teachers" of mathematics, who portray the subject as consisting of ineffable, uncontestable rule-sets... It is surely fairly futile, but I think any approach that does not tell kids to "trust, but maybe refine/improve, their intuition", is doomed to disengagement.
The same appears to be true of grad students in mathematics at good universities, etc., btw.
So, yes, I'm resisting answering the question as asked... but eminently willing to edit, considering that I do think this is an important sort of question.