No, I did not "have that backwards". I didn't bother to mention their names prior to leaving Earth 2 (no dash between "Earth" and "2" with the New52 version, at least not for the title of the comic book), and whether they could keep their code names from their time on the "main" DC Earth doesn't matter. I had thought Power Girl had changed back to going by "Supergirl" or maybe "Superwoman" since she had a very prominent "S" crest in Earth 2: Society, but I never read that series. I did hear that Helena dropped her "Huntress" name and now wanted to be known as "BatMAN.
Geoff Johns (for one) is very bad at writing certain characters. Power Girl appears to be one of them.