Well epi 1 is much earlier then 3 and that is when she decided to kill her self as she didn't want to be an assassin any more. Angel Heart herself decided she didn't want to kill anymore.
Killing ppl just coz she feels like it isn't the City Hunter way, they are far far more subtle then Revvy. Since you have decided to give up you won't see how much influence Ryo/City Hunter has and how he uses it to resolve situations that don't need to be solved with a bullet in the head.
Even in CH a lot of the scenarios were not solved with killing, never mind mindless killing....
Both Revvy and Shang Ying's childhoods are messed up, but Shang Yin got conditioned to be a professional killer and she decided to end that way of life the only way she knows how. Revvy on the other hand is psychotic on the level of lusipher from poison elves, both she and luse have the same type of childhoods and both ended up with the same type of outlook to life and the same solutions to their problems. See problem, kill it. Shang Yin stalks her prey and analises the situation and then uses her training to solve it. Unfortunately that training is something she doesn't want to do anymore. Kaori and Ryo like any good parents are giving her more options and skills to deal with these encounters, especially normal day to day ones.
When Shang Yin was reborn when she gave up on being Angel Heart, she was reborn as Kaori and Ryo's daughter. Personally I think Kaori couldn't stop her killing if Shang Yin really wanted to kill, but then the heart would reject the body which would be a bad thing...
To be honest if it wasn't for the extra info we get as viewers, we could say that the Kaori factor was just the way Shang Yin's conscience materialises itself for her.
Anyway as a big fan of both Kenshin and AH I can safely say they are nothing alike
