I'll attempt to answer your question in reverse order.
What would I do in a similar situation?
The choices for what we now consider standard definitions were refined countless times before the a single definition was accepted. It is easy to forget that and it is partly our job as educators to explain this from time to time to the students (least they start thinking mathematics is a static discipline). So, such situations were students show a distrust in a particular definition that we know is a really good one is an excellent opportunity to engage the students with the dynamic nature of mathematics. I usually encourage the students then to scrutinize the definition, making their objections explicit. Depending on how much time I wish to spend on such things I then decide whether or not to look into their objections more carefully or not.
How do I handle this situation?
The two particular difficulties you mention I would treat as follows. Difficulties on whether or not $f(x)=\frac{1}{x}$ is continuous are the result of being careless with specifying the domain. This is thus an excellent opportunity to explain that the question "is $f(x)=\frac{1}{x}$ is continuous" is meaningless until one specifies the domain. If there are some books that fail to do this properly, then it's a good opportunity to warn the students of lousy books.
The second situation, that of vacuous situations, can indeed be quite confusing but I would not spend too much time on it since such things rarely show up in nature. Moreover, once students are a bit more proficient with mathematical arguments they will easy dispense with such trivialities themselves, so the problem solves itself. If students are really troubled by this I would tell them to spend 15 minutes thinking about it and resolve it themselves.
Can I do better?
Well, if one can no longer improve, then one is dead, so yes, you can also do better :)
The particular notion of continuity is actually a very tricky one, at least historically. Our intuition for what continuity means is quite bad. It is common to say that a function is continuous if its graph can be drawn without lifting the pen from the paper. This is wrong of course, as it more accurately describes functions that are infinitely differentiable at all but finitely many points. It is actually quite difficult to geometrically understand continuous functions. This is something the students should be able to relate to - after all it is difficult for them to understand this concept. Also, historically, mathematicians struggled long and hard with this concepts with many disagreements between Lagrange, Cauchy, Bolazano, and Weierstrass. Some of these historical accounts can also be conveyed to the students. It can also be mentioned that the standard Cauchy definition is not the only way to formalize the notion of continuity, just to tell the students that more than one road exists.