Vitter's bill is about "birth tourism" and anchor babies. It doesn't address presidential eligibility but common birthright citizenship. That is citizenship derived from birth parent's citizenship and/or birth location.
In California we had a clinic aimed at Chinese nationals specifically for them to come, give birth and go home. They did this so the new child would be American and they could avoid China's "one child" policy.
If you need the problem of anchor babies explained, read the article.
We have doctors in Nogales, AZ that have set up advertisements for such “births”. Citizenship whether Naturalized or Birth on soil is one thing, but NBC, those words chosen by the Constitutional Authors is another.