She's not. While I'm sure Cyclops would be glad to have her, he also respects that his old friend wants to maintain a secret identity. That'll be why...
Both work. Spider-Woman was the name used in the Ultimate cartoon, but think about the back-up story from last week's Giant-Size, which was a reprint...
Like how we got a Supergirl show where we only saw glimpses of Superman before he appeared for real and then later he got his own spin off show that...