No matter what word the Vatican used, this is fundamentally about trademarks, not copyrights.
Trademarks are about identity: they prevent impersonation. Identity protection is exactly the Vatican's concern here. They don't want other groups pretending to be the Catholic Church. (There remain interesting questions as to who has the right to decide what the Catholic Church is, as with any religion or other affinity group, but that's an inherent property of identity itself.)
This case is not about copyright, in any case, since copyright isn't about identity. When people illegally share music, for example, they don't remove the original artist's name and replace it with their own. Copyright is not about credit or attribution; it's about the sharing itself. That's why it's called "copyright" instead of "creditright".
Lumping these two unrelated concepts together under the term "intellectual property" just leads to confusion. And the copyright lobby is very happy to benefit from this confusion, since people have a strong moral attachment to accurate crediting. For example, see http://questioncopyright.org/promise#plagiarism-vs-copying for a blatant example of the RIAA trying to confuse copyright violation with plagiarism -- that is, confuse unauthorized sharing with identity theft.
Now, if the Vatican were claiming a monopoly right on the Bible, that would be a copyright issue. But they're not, of course, because the Bible is in the public domain everywhere.