Re: Did Supergirl get punched out of her skin?
I think it was the Superman animated series (and later JLA) which sort of mixed the original concept with the look of the Linda Danvers version. They of course eventually brought back a post-crisis version of the original Kara (In Batman/Superman) who has become a staple of the post-crisis Superman mythos since, although with the relaunch I think her origin has been altered (Along with Supes himself). I think in Justice League Unlimited, the DCAU version of Kara also started wearing an outfit similar to her comics counterpart.
You've actually got that backwards. The white belly shirt version of the costume originated with the JLU cartoon and was introduced to the comic later on.
Baby, you and me were never meant to be, just maybe think of me once in a while...