I agree with this
"That's when she realized she was unable to love him. She had to let him go because she dosent have her true personality (kind, sweet and loving) anymore. "
BlueberryLove
Everyone is different, and so is Yumi. She broke the moral she once held (not dating someone who easily gets swayed) just to try dating Babi again. It's not an easy thing. That's the moment she lost herself.
In this drama, we might only see her feeling numb when seeing Daeun's call. But in reality, falling out of love is....traumatizing. Knowing that you can no longer love the best person you choose to be with you. At that point, she can't even trust Babi anymore, she has nothing to expect, she's basically losing her purpose of being with Babi. As annoying as it might seem for us, it must feel even more frustrating for her. Of course, there's no guarantee that she wouldn't like, accept, and love Babi anymore. But there's no guarantee that it would work either. If continuing the relationship means preparing for marriage with doubt in mind and uncomfortable feelings in her heart, then it's too much. She must let it go.