Your template settings uses a global color for both of these text colors, hence why it changes on both sides.
The only way to change this, is to add CSS in your styles.css file that overwrites these particular instances. If you post the website, I can write you the code you'd need to add.
This is an accepted solution.
I went ahead and added these settings for you in the theme to specifically target the cart and search bar placeholder text color. You can assign a new global color value in your settings under Theme Settings > Color > Text > Search and Input Text
Hope that helps