Board Thread:General Discussion/@comment-5085906-20130627191357/@comment-5085906-20140621104744

Raiden was called Snake at the beginning because his superiors decided to give him that codename, whereas in Big Boss' case, there's no one higher than him in his army so he can choose his own names.

I think this Punished Snake/Venom Snake thing is more of a nickname - something to describe Snake's inner emotional clusterfuck and his eventual descent into villainy, similar to how Old Snake described Solid Snake's physical appearance.