I'll just put your answer here because it made "click" in my head when I read it :
If you’re using a blackbody for a light colour it should be because you’re trying to simulate an incandescent of that temperature.
Blackbody colours have precisely nothing to do with the colour of the sky. As others have mentioned the fact that literature refers to colours as “sky blue” is an attempt at classification.
Thanks again everyone, you rock !