This is looking like a promising adaptation. I was concerned about some aspects of the teaser, like the way they repurposed the context of the, "I'll give you my heart" line, but it looks like this movie will be covering a decent chunk of Alita's character development.
When the movie was delayed, I suspected they might re-design Alita, given the overwhelmingly vocal reaction people had to how distracting Alita's huge anime eyes are. Looking at the following shot side-by-side, it's clear that they have re-done Alita's look at bit (and changed the colour timing of shots), but they've gone the opposite direction from what I expected. I thought they would make her more realistic, but instead they've rendered her in a softer, even more anime-esque style, probably with the intent of pushing her out the opposite side of the uncanny valley. It's not easy to tell the differences side-by-side, but I've tried overlaying these images and flicking back and forth, and that's definitely a new render. The hair is different, the irises of her eyes are larger and less glassy, and her skin texture is different.

Somehow it works. By making her more cartoony, she stands out less.
Everything I'm seeing in the new trailer looks promising, and suggests that Rodriguez and Cameron have a good handle on the important aspects of the characters.