Characters don't concede or accept concessions. Only players do that.
Concession, as a mechanic, specifically faces outward to the real people playing. It's explicitly about the parsing of narrative authority over the fate of the character post-conflict, and nothing else.
Characters do not concede. People do.
-- Fate SRD, "Conceding the Conflict"
So, Dangeresque and Perducci are out for each other's blood, having a knock-down drag-out fight in the bowels of the ship.
Perducci shoots Dangeresque in the gut with a harpoon gun, pinning him to the hull and punching through a weak point. As the ship lists, taking on water, Perducci escapes with his life rather than confirm his old enemy's death.
This is the result of a negotiated concession between a player and the GM. It is not, and I need to stress this, the result of Dangeresque turning to Perducci and saying "hey man, you're kicking my ass real bad here and I'm not up for dying right now, so how about instead...". That would be comical. Farcical, even!
The players of the game are coming together to make a decision about what happens next that temporarily sets aside apparent character motivations. You'll see this even in the example conflict in the Fate Core book. When it's time to pick sides, it's the fairly pithy and straightforward "Og and his buddies want to do in the PCs", but when Landon concedes? "[Og] knocks you out, spits on you, and takes your sword."
Characters don't really pick an absolute, immutable goal on their way into the conflict, so there can be some flexibility in how they exit it, in success or failure.
"But my character would accept nothing less than your character's broken, lifeless body/reputation!"
Easy there, cowboy. Your character is not an immutable being separate from yourself. That way lies My Guy syndrome, and that's not a good time for anybody, even you.
You get what you want out of the conflict because you won it, but crucially, there wasn't a time where you made a secret note before the conflict about what you wanted, and now its contents are binding for everybody. There also wasn't a time when you made an open note before the conflict about what you wanted and everybody agreed to it, and now its contents are binding for everybody. There are systems where you do pre-set stakes, like Shock and pretty much the entire Burning Wheel family, but Fate isn't one of them.
When a concession happens, or when one side takes the other side out, then it's time for you, as a player, to decide what your character will accept. You do make informal statements prior to this, usually as part of taking sides for the conflict, so that everybody knows the rough scale they're heading into, as suddenly finding out you've been in a bake-off TO THE DEATH often offends. But now is when you decide.
And if somebody's conceded out of the conflict, they also get to decide what they'll accept happening to their character. That's the benefit of conceding instead of hanging in until you're taken out.
Everyone Writing A Story For Everyone
Fate is a game that's a little more explicit about how you're both the player of a character and a contributing author of an interesting story that involves everyone's characters. The entire Fate Point economy, particularly the part of it that's compels, makes use of that author stance - as the author you're aware that your character is going to make things harder on themselves and the payoff is a point for you. As a character you probably don't think you're doing anything particularly awful but then things suddenly turn south.
Concluding conflicts involves working together with your fellow players as authors to mark out a little space for the concessions and plot a course for the winners. So all you really need to do to stop this resolution from going for maximalist stakes is to find literally anything at all that the winners want more than to end the losers' story. And there's an entire universe full of things to have, some of which probably make a lot more sense to take now that you've proven yourself superior to the person who previously had them.
Just look at Og. It turns out they wanted to take a trophy more than they wanted to be a cold-blooded murderer.
