Changing light’s color temp vs changing camera’s color temp

I’m curious on how DPs choose which (lights or camera) to adjust the color temp for.

I know movies like Thirteen become really blue at some point, and that probably required the camera’s color temp to be adjusted, but what about in a controlled situation like in a house. If you wanted a moodier, cool tone, would you make the lights cooler and keep the camera the same or the camera cooler and keep the lights the same?