I agree that it is difficult to communicate the issue of well-definedness.
First, obviously, in real life there is nothing like this. We rarely accidentally make a reference to a thing that is non-existent (well, ok, let's ignore politics for a moment).
But, really, people have absolutely no experience with this potential issue. And, in fact, I would claim that (by this year) a mathematical discussion that includes a requirement of proving something is "well-defined" has prior epistemological and ontological problems.
Yes, this is a bit of a "peeve". I claim that the issue should be about existence of a (often "universal" in some way...) object... as opposed to writing formulas that may or may not make sense. Yes, I understand that a certain philosophical viewpoint retreats (!?) to only claiming that well-formed formulas are all we should contemplate.
Yes, I'd claim that students should rarely be asked to "prove that X is well-defined". They have no idea, and discovering that fact proves nothing. The more genuine question, in my opinion, is whether the alleged X is sensible or not. Or is it random and pranky? For math people, I think we really should recover from the random-pranky lifestyle... supposedly (but dubiously) teaching students about logical precision.
Quite seriously, what I try to communicate to students is that mere formulaic stuff is not what we want. Writing strings of symbols guarantees nothing. There should be a meaning behind the narrative. In that context, what the heck could "well-definedness" possibly be? Seriously! It is a bit amusing that mathematics has arrived in a situation where "well-definedness" should be a real issue.
By this year, I try to insist that the math grad students respond in a different way to such questions... In particular, that the notion of "well-definedness" is generally deficient, if lacking a larger context. Yes, I do claim that the notion itself is intrinsically problemmatical. Not that it won't arise in certain formalized situations, but that it doesn't capture anything truly useful.