There's really no defining point. Many bands who were more obvious about their faith in the past, and are now more vague are still labeled Christian by many, as long as their lyrics don't begin to reflect anything against the faith. But some people will stop labeling them Christian because their lyrics don't reflect it. Its really down to what your personal opinion is that qualifies as a band as Christian. There is no specific rule where once you cross a certain line you can no longer be considered Christian.
I was a fan of POD back when they released Snuff The Punk, and even into some of their major label dealings. But After they got signed i did struggle more and more with where they were coming from. Eventually the music lost my interest as they ventured soooo far away musically from where they started. And with all the weirdness going on about their beliefs, i just couldn't continue listening.
They now seem rather bold with their Whosoever stuff, and have pretty credible artists backing that, but i'm still leery of the band. And the recent revelation that they have long desired to use cussing in their albums, which indicates a bit into their persona lives, i'm just done with them all together.
Far as recommending anything, well, if Skillet and Relient K are your starting points, then i have nothing. Not really into those styles. But i am curious why you are looking for neutral bands specifically. Why not more Christian bands as well?