Teaching undergrads and graduate students, by this point I do not like "precise definitions" at the outset, for the sort of reasons given by W. Thurston, and also for reasons given by I.M. Gelfand. Namely, a suite of examples of phenomena and of causal mechanisms to illustrate various aspects of the topic at hand. Arguably, only after looking through the immediate examples can we understand what our terminology needs to cope with and refer to.
Also, I've found that most undergrads have cognitive or semantic troubles with "definitions"... since very little else in the world works this way. They often react by "memorizing" the definitions, rather than integrating them with phenomena. Artifacts of formatting and wording are often indistinguishable to them from genuine issues. This seems bad to me.
By the time students are in grad school, they seem to be "reconciled to" checking definitions without questioning the virtues of them. But, indeed, just as with dictionaries of human languages, definitions of complicated things are often not infinitely precise, and meanings or references drift with time.
The notion of "function" exemplifies some of these troubles. First, it would be silly to deny that at entry level most functions are given by formulas. Second, yes, we want a notion of "function" that does not depend entirely on formulas. Third, yes, the set-theoretic epiphany to define a function $f:X\to Y$ as a subset of the cartesian product $X\times Y$ with certain properties was a very nice umbrella. "(E.g., Lebesgue) measurable" as a qualification was/is very useful,... and/but leads to a slight loss of pointwise sense, since now "almost everywhere" turns out to be the useful degree of precision. And, indeed, $L^2$ spaces of "functions" do not have pointwise values, yet are the right things to look at for many purposes. Similarly, "generalized functions" (a.k.a. "distributions") usually do not have pointwise values, so are certainly not specifiable by "graphs".
... though, yes, the Schwartz Kernel theorem does recover many of the lost items!
And so on.
In particular, "pointwise convergence" is not usually the real issue, to my mind! Convergence in $L^2$, or in Sobolev spaces, or uniformly pointwise, sure.
So, then, one of my first talking points in teaching graduate-level real analysis this year, with an eye to updating the course content, is to illustrate the various troubles with "pointwise" notions, arguing that we need/want a broader outlook on the notion of "function", at the very least to be able to accommodate actual practice!
From the viewpoint of exposition, too, I do not like definitions. If/when they appear without motivation or examples, why should I accept them? Why should the students? An implicit justification "by authority" is not the kind of reason I want to promote. I want to see phenomena first, and any discussion of modifications of word-usage afterward.
That is, although I realize that quite a few people like the other-worldliness of mathematics, a great many students have difficulties, and I think with good reason, operating in a context where they are essentially required to suspend their own judgement and play by rules whose significance and origin is effectively kept secret. There may be virtues in the formal exercise of playing such games, but it is not clear to me that this is really the main selling point of mathematics.
Yes, I do like actual proofs, rather than merely heuristics, and this does require precise references and referents, but that's not the same thing as "definitions". Also, proving boring, superficial things very precisely is an odd sort of exercise, of limited interest to many people, or only of interest at very particular moments. A more interesting kind of thing is, for example, legitimizing the mathematical aspects of things physicists (Dirac, Bethe, et al) have done, with observable physical corroboration, to repurpose the mathematics more broadly. In that sort of circumstance, it is not within our power to "define" anything into or out of existence. Instead, to my perception, the issue is of grappling with examples and phenomena. Yes, there is the secondary issue of mutual intelligibility, but it does seem to me secondary.