As you pointed out, Lara's first kill in the 2013 game kind of falls flat for me because she immediately goes on to kill so many more people and doesn't even blink. And then there's the part where she falls into a cave that is filled with blood and bodies, and it doesn't faze her in the slightest. She swims through the blood like it's water. This is the same girl that was getting choked up over killing someone who was going to kill--and quite possibly rape--her, right?
Honestly, I think when she finally kills Larson in the remake was more impressive in the way they handled it. She struggled with the decision, and even at the end of it, she's still horrified with what she did. The only thing that kind of ruins it is when she wipes the non-existent blood off of her hands. Since there's no blood on her hands, it looks weird.