You've somehow got a core component of both Laura and Logan's stories and personalities entirely backwards.
The Weapon X program tried to turn them both into living weapons. Both of them fought against that, and became their own individuals. X-23 wasn't trying to avoid being Wolverine. She was trying to avoid being the weapon that Logan also has avoided becoming. She wanted some separation from Logan because, having grown up in such a controlled environment, she wanted to be completely free to find herself, without her clone-dad providing a constant pattern for her to model herself after.
But she got past that. That's called "character development".