And Now The Rest Of The Quotation
"worse outcome, hard bargain, or ugly choice" is language quoted from Dungeon World's parent game, Apocalypse World. Specifically from the downside to the move Do Something Under Fire, or Act Under Fire depending on whether you're looking at first or second edition.
Dungeon World tends to fall a little short in the area of worked examples, so I'll put up what you might consider canonical examples of each of the three, from pp.136-138 of AW second edition, and explain how they differ.
Keep in mind that, whatever the 7-9 is offering you, it's a qualified success. It's not the worst thing that could happen.
The Worse Outcome: Almost Got 'Im
Bran the savvyhead's got less than a minute to get Frankie's car started
again before Balls and friends are on them. (On a 7–9, maybe I give him a
worse outcome: he gets the car started, but Balls' first couple of people are
There isn't any choice to the worse outcome. It's the "any color car as long as it's black" sort of offer. I mean, what, you're going to just stick around and let everyone catch up? No.
It's often used in situations where you've got a bright clear picture of one single thing that could go wrong and the best thing the PCs can hope for is not to have anything go wrong.
The Hard Bargain: Something To Lose
Keeler the gunlugger's taken oﬀ her shoes and she's sneaking into
Dremmer's camp[...]. (On a 7–9, maybe I give her a hard bargain: she can get to Dremmer, sure, but only by breaking cover and alerting his guards.)
The hard bargain is giving up something to get something else. It is a choice - do you want to give up your stealth to get to Dremmer, right now?
Left unspoken here is what happens if you don't take it, but it's clear enough that, well, you're in a position to take it. You know how to get to Dremmer. You know you're hidden and going this way is going to take you out of cover. Okay fine, you don't go that way, so what are you doing?
(What you're doing probably involves somebody making a move and making a roll and perhaps a snake-eyes happens and you break stealth anyway, but it's still yours to risk.)
The Ugly Choice: Nothing To Gain
Foster's gang opens fire on Marie the brainer and Roark her friend
picnicking on the burn. Roark's hit, and Marie tries to drag him to cover.
(On a 7–9, maybe I give her an ugly choice: there’s a second bullet, and will
she take it or will Roark?)
The ugly choice is the hard bargain's meaner brother. You can't turn it down and leave yourself in a neutral position. The hammer's going to fall. You just get to pick which sacrifice you can best deal with.
Aren't you the fortunate one?
And Now You Know The Rest Of The Story
So, let's circle back to your example, which, having seen the difference, we can classify as an ugly choice. ...probably.
I mean, the worst thing that could happen was that the PC ate a bunch of damage and let go of the rope, right? So "take a bunch of damage" and "let go of the rope" are both things the PC doesn't want to happen, there isn't a single worse-but-not-worst outcome, so instead of both happening, they get to pick one.
Or it's possible that this was a hard bargain and the rope represented some kind of advantage in progress. Like the PC was trying to lasso an electric owlbear or whatever and you're like "it shocks you and you drop the rope" and they're like "what no I'm a lasso that owlbear right now", in which case the bargain is, they can give up some hit points to get the rope around the owlbear.
It... might potentially even have been a worse outcome? Hard to think how. I am assuming in this case that the rope had been, like, tied to something important with time and care, and wasn't just "okay, I spend an adventuring gear and now I have some rope." "You lose hold of the rope." "Okay, I spend another adventuring gear and now I have some more rope." In that case losing one use of adventuring gear is probably strictly superior to losing some hit points, but that's just a probably. When it comes down to it, they're really not the same thing on the same scale.
Whatever the circumstances, the objecting PC felt that there were multiple possible vectors of badness they might have been able to choose from, and you made that choice for them. You told them the bad thing that happened. That's not really any kind of a success, and they were right to object.
But Which, And When?
Well, first off, you're the GM, alright? You control the universe. You can pick whichever one of these speaks to you the most at the time, and make the call, and that's what happens.
If things don't seem to you that there really was a choice in the matter, if it wasn't really possible to hold onto the rope when the electric owlbear shocked you because it's just that potent, then the next best thing to a clear success is dropping the rope and not getting hurt, and you're not obliged to offer a choice. (It is, however, proper sportsmanship to listen seriously to player objections and explain yourself. Who knows? Maybe you'll find yourself coming around to a second Defy Danger to grit your teeth and hold onto the rope, with damage guaranteed and risking a debility, because if that's what an electric owlbear can do then they really want to lasso it now.)
If you some more guidance than just that, well, Defy Danger is kind of a catch-all move, right? The book actively steers you away from penalizing the roll if the PCs are in more dire straits. There is a danger, you wish to defy it, so roll. But the circumstances are still what they are.
A hard bargain best suits a controlled position. The PCs have prepared for this and they're facing it head on, and you're offering them a way to escalate things but on their terms.
A worse outcome best suits a risky position. Things have started to jump the rails a bit but not beyond the usual ability of the PCs to deal with it. They can't dodge everything but they take less damage. They can't avoid all the attention but they do avoid most of it. They don't lose everything but they do take their pick of what to keep.
An ugly choice best suits a desperate position. Oh God, oh God, please let this work, oh God, please please please let this work, oh God. Unless by some miracle you get away clean, the best you can hope for is to choose what takes the consequences.
(The astute observer will have noticed what else I'm cribbing this from, and infer that in advanced cases the GM may offer the PCs some benefit to drop down to "worse outcome" or "ugly choice" from a better position than would warrant it. They are both astute and correct.)