Translation:I want to break up with him.
"I would like to break up with him." seems like an answer that should be accepted given that "想" has been treated as "would like" to differentiate it from "want" in previous exercises.
"I'm thinking of breaking up with him" was rejected but I think it's ok
I wouldn't understand 想 as "to think about" in sentences like this. I would translate your sentence as 我在考虑跟他分手。
No problem, I'm native speaker. I agree 我想跟他分手 = 我考慮跟他分手. These in Chinese logic are interchangeble. If you say 我想要跟他分手, it's " I want to break up with him."
我想"著"跟他分手 is wrong. Though 想著 contains the meaning of think about, but it can't be used everytime. There's some certain condition.
我想著這件事 I'm thinking about this problem.
我想著你 I miss you.
Yeah, 想着 can be interpreted differently in different sentences.
Regarding 想 and 想要, I don't see they make any big difference in this sentence. I would rather distinguish between 想(要) and 要. I understand that 想 expresses the desire and even a pending action, but I never think 想分手=考虑分手 except for examples like: “想分手有用吗？不如想发财。” — “Thinking about splitting-up is useful? It would be better to think about getting rich.”
Correct me if I'm wrong, but I think 想 means "to want" and "to want" doesn't automatically imply "to be going to".
Here, 想 means consider. It's neither want nor be going to.
我要跟他分手 I'm going to... 我想跟他分手了 I want to...