Self vs. non-self (and the integration of the Winter Soldier into Bucky Barnes)
As you pointed out in your question, my Bucky has incorporated the Winter Soldier into her self-identity. Whereas MCU!Bucky says, "I am not the Winter Soldier anymore," my Bucky, like 616!Bucky, has accepted that the Winter Soldier will always be a part of her. She's discussed this before with @invncibleiron's Tony -- that Bucky Barnes and the Winter Soldier aren't two separate personalities.
Bucky has come to the conclusion, looking back on her life, that the seeds of the Winter Soldier were always there. She's always had a certain ruthlessness to her, especially in defense of the people that she cares about. As a child in Brooklyn, that usually involved not being afraid to use her fists to teach someone who hurt her sisters or Steve or her friend Rosie Finn a lesson. She fought by trying to get the bullies on the ground or running away as quickly as possible, even if that meant breaking a nose or knocking out some teeth.
These people are mine. How dare you try to harm them?
Still, when she was young she did her best not to leave wounds that wouldn't heal. But the brutality of the war and the trauma from having been raped, tortured, and experimented on in the Red Skull's factory stripped away a lot of her younger self's qualms about that. She found that killing came easily to her, and also found that she didn't feel particularly bad about killing, either. It had to be done, so she did it -- competently, thoroughly, and without remorse. The Nazis and the Red Skull were murdering innocent people and causing untold suffering for those who survived. Bucky found that unconscionable, so she did everything she could to put a stop to it in the most permanent way possible. Dead Nazis can't kill again.
This world is mine. How dare you try to destroy it?
The pieces of Bucky that would comprise the Winter Soldier were already in place by the time she fell off that train. All that was left for Zola and Karpov to do was to strip away the parts of her that weren't the Winter Soldier. The Winter Soldier is just Bucky Barnes with her conscience, memory, and free will removed. The fact that she was already suffering from amnesia caused by the fall and her subsequent death made their job easier. It would have taken a lot more work to break a Bucky who could remember who she was and what she believed.
I think it's important to point out that, even as the Winter Soldier, she wasn't a bully. She was competent and brutally efficient, but when she was controlled by the Soviets she always made an effort to avoid collateral damage, and she clashed with her handlers over that tendency on more than one occasion. Alexander Pierce broke her of that habit, but even when she was under his control, she never took any joy in hurting people. She followed orders. She did the job that was assigned to her, because that was all she knew how to do, but she did it without passion or pleasure.
When she finally escaped from HYDRA and came back to herself, she had to come to terms with the fact that yes, she had been the Winter Soldier. More than that, she still was the Winter Soldier. So she decided to make use of that fact. Bucky has always been practical. She'll deploy any tool in her arsenal, and the Winter Soldier is now another tool that she can use. People are terrified of the Winter Soldier, so she'll use the name and the reputation to strike fear into the people she has to fight.
There's also one additional fact that Bucky knows but has never given voice to, which is this: She's seen some of the worst the world has to offer, humanity at its cruelest, most wretched, and most malicious. Based on those experiences, she's come to the conclusion that sometimes, to make the world better and to spare innocents from death and suffering, certain people have to die. And that being the case, she'll take the responsibility for killing those people, because she can bear it. She knows some of her friends, particularly Steve, would cut themselves up inside over having to make that call. So she'll make it instead to protect them.
