For every encounter, start by asking: what's really at stake? What do the PCs get if they win, and what do they lose if they fail?
Something that would end the story on failure is never a good stake. Usually, this includes "the lives of the player characters", outside of a deliberately lethal conflict. If that's really what's on the line, you should rethink what the PCs are fighting for. Wherever possible, the stakes should reinforce the story being told, and they should be appropriate to the genre of your story.
If Superman is rescuing a crashing jetliner in his own comic book, and Lois is on it, then you can be pretty sure that Lois Lane isn't going to just die, because that will be jarring for the readers of a Superman comic book. Something else must be at risk. If Lois was injured badly enough in the rescue, would Superman be distracted from something more important? Would a villain be able to advance his plan further if the Man of Steel was in the hospital sitting by Lois' bed? Balancing his responsibility with his humanity is a core part of Superman's story, and those sorts of stakes reinforce his nature.
Sometimes, the stakes only become apparent as the encounter resolves itself. In "Blade Runner", during the final fight between Deckard and Roy Batty, Roy is already dying, and he knows it. Over the course of the conflict, the real stakes are eventually revealed: does his life matter? Will he leave behind any sort of legacy? Deckard is the last chance he has to make that happen. The stakes for Deckard seem to be his own survival, but they shift to protecting his identity. Can he keep being who he was? Does he want to?
If your group realizes that something has changed, it's perfectly legitimate to stop the currently running Challenge, Contest, or Conflict, or to reframe it in the new terms. In the Deckard vs. Batty Example Fight from Richard Bellingham (discussion thread here), this is exactly what happens, as the physical conflict becomes a mental one.
In "Indiana Jones and the Temple of Doom", Indy, Willie Scott, and Short Round are riding the rails in mine carts. Their goal, of course, was to avoid the villain's henchmen and get out of the mines safely. Shorty was explicitly told which route to take.
Did they go that way? Of course not!
Did failing to go the right way end the story? Of course not.
A good encounter works like a mine cart ride: no matter where you go, you have to keep moving. Like mine carts, a story is always worse if it derails than if it takes a new route. Don't fear failure - instead, fear not having a new direction to offer the group if the dice don't work out.
So what should you use - a Challenge, a Contest, or a Conflict - to model an encounter? For me, it depends on the ways the participants can leave it. Do these exit points center around the stakes (Contests), the participants (Conflicts), or the situation (Challenges)?
Contests have twists and turns, as rivals try to outmaneuver each other, but you're all still heading toward something or away from something. Everyone exits the encounter when one or another got where they wanted to go.
In a Conflict, exit points always come at a cost. Nobody makes it out in the same shape they came in, unless they bought their victory with hard work, effort, and good fortune. Even concessions carry a price.
In a Challenge, there's many ways to exit the encounter, with the PCs winnowing down the possibilities. Sometimes they get the outcome they most want. Sometimes they compromise. Sometimes they're left with a terrible choice. But the exit point they get to use is one they can afford with their resources of skill, Fate points, and luck.
Challenges, Contests, and Conflicts are tools. They offer a way to challenge your players and provide a believable, if sometimes surprising, outcome. Once you have a sense of what the PCs are risking, and an idea of how their actions will carry them from scene to scene, you will be able to choose the right tool for the right job.