My interpretation is this:
Goddess' Sword
- the actual physical sword is the Goddess' Sword, as in the sword used to belong to the goddess, or is related to the goddess somehow (wait, wasn't the goddess holding a sword in the opening scene?
). The fact that the name has a possessive term on it (Goddess') makes me think it is only the physical blade. Why would anyone actually own a spirit (Phi), and take possession over it?
Skyward Sword
- I think it includes both the Goddess' Sword and Phi because of the two meanings confirmed by Nintendo on the name: Skyward meaning "to the sky"; and Skyward meaning "sky protector". The second one is what makes me think that the term "Skyward Sword" also includes Phi, because Phi, as a spirit, is in a way, a kind of protector.
PS: Why does the Vatican care about Nintendo so much? How much power over the people do they think Nintendo has? Are they afraid Nintendo will spawn a new religion against them similar to how the protestants protested against the Roman Catholic Church?