For me, a saving throw is one last chance to avoid really bad consequences from a situation. A couple other bloggers have mentioned the same belief. A character drinks from a fountain only to discover it's poison; the bad decision has already been made, but the saving throw is one last shot to avoid death.
But I go a little farther than that, in two ways. One is that I distinguish between magical/unnatural situations and mundane but deadly situations. With one exception, the standard saving throw categories all have to do with fantastic dangers: magical wands and spells, the gaze of a medusa, the breath of a dragon, a cursed scroll of polymorph. The exception is deadly poison, but one could argue that instant death poison is extreme enough it's worth counting it as fantastic. For non-fantastic dangers, like falling into a pit, the rules mention a simple d6 roll. The difference is that fantastic dangers become easier to shake off as characters increase in level, but natural dangers are a flat 2 in 6 (or whatever the roll is.) I believe in maintaining that distinction, so I would never apply one of the standard saves to something like crossing a tightrope; instead, I'd make a situation roll to see if the character slips.
The second way that saves in my games are different from saves in some other people's games is that I believe they should be the result of a last-chance action. You drink a potion, I say "it's poison!" you say "I spit it out!" This is worth a saving roll. I actually haven't been a real hard-ass on this so far, giving players a save for some things like poison even when they take no special action. I rationalize that some things, like drinking poison, might trigger an automatic physiological response, like vomiting. But save vs. dragon breath is supposed to represent ducking and covering, and save vs. petrification is supposed to be the character shutting their eyes and averting their gaze. If there's some reason a character would be unable to take effective action, there is no save.
The side effect of this way of thinking is that I will allow bonuses to the roll or other benefits if your last-chance action is exceptionally good. You drink an ipecac immediately after that bottle of poison? That might be worth a +1 or +2. You dive into a pond when the dragon breathes fire at you? Half damage even if you fail your save, no damage if you succeed. You were already under water before the round when the dragon breathed? No roll necessary.