A penetrating question, in my opinion! The use of "formal" in contemporary (1980-2012) mathematics is very volatile and brittle, in fact, in all my experience. It is very reasonable for a cautious person to feel uneasy. :)
One unfortunate use equates "formal" with "the symbols seem to give this, but I can't justify doing it, ... but no one who cares about the outcome cares about the justification, so we'll just not worry, and no one else will, either." I know, it is possible to rationalize this, but it's not a good thing, in the same sense that I tell beginners to think carefully when they say "clearly/obviously", because either they should be able to give the one-or-two-line proof, or it's not truly sooo easy. That is, what with human foibles, we have the "it's so obvious I can't explain it" problem.
At the other end, "formal" can mean that everything is set up so well that we need merely execute some (already fully legitimized) symbol manipulation to obtain a result. Unnervingly, this usage is often mixed with the "we can't justify this, but the symbols seem to work" (=heuristic) usage.
The unfortunate example of "formal power series" adds another usage: as is well known, here "formal" means merely "not convergent", and/but this resonates with a slightly misguided sense of the "we can't justify it, but the symbols look ok". In fact, "formal power series" have a sense and meaning that does not depend on "convergence", so is not attenuated by their non-convergence. By this year, it's not "just a heuristic", but is fully legitimate, ... although 250 years ago many people (e.g., Euler) saw that the arguments gave interesting outcomes, without worrying about (being able to give?) justifications.
The further muddle seems to be that, of course, an excellent heuristic is more interesting, more productive, than many fussy proofs, so we don't object to heuristics by any means. Thus, we do not object _in_principle_ to non-rigorous interesting discussions. But our mythological commitment to "rigor" puts heuristics in an odd limbo state: with luck, a non-rigourous heuristic can conjecturally explain otherwise-unintelligible phenomena, but, also, (let's say) "unfeeling" extrapolations of otherwise-sensible ideas seem pointless.
One problem is that it is very difficult to "formally" (!?!) distinguish the two, that is, productive heuristics from superficial-and-silly heuristics.
Much more can be said... but, at least, yes, a perception of inclarity is more-than-reasonable. There is not a consistent use of "formal". Examination of contemporary use of this term is professionaly-sociologically quite interesting, but does not explain any particular case. Rather, in summary, the best contemporary use seems to be "here is a slightly vague idea, which seems to be useful, but there are missing pieces... nevertheless, I will put it into symbols for you..."
(All the more hilarious given the popular conceit that mathematics is oh-so-rigorous...)
Edit/addition: given the inhomogeneity of the audience, perhaps it is necessary to note that a "formal power series ring" $k[[x]]$ is a projective limit of quotients $k[x]/x^N$, so that it is a legitimate object... but/and while the notation for a given "formal power series" is indistinguishable from the notation for convergent power series, there is no implication that formal power series do or do not converge. Yes, there is a natural injection of convergent power series if $k$ is a valued field, etc.
Also, I would reiterate the comment that, while "rigor" is often desirable, if not too expensive given the goal, for example, it is not the only goal, surely. Further, "rigor" is surely relative. And, if we distinguish the process of learning/doing mathematics from the edifice itself, surely there is as much reason to find motivation and heuristics to discover/explain important phenomena, as there is to be as-scrupulous-as-possible about small or irrelevant or uninteresting things?