If you keep checking on the references to this site's entry for
Camille, it will eventually take you to the original Latin form
Camillus, which it says
may have meant "attendant at a religious service," though the name may also actually have an Etruscan origin and an unknown meaning. If you think back to the fact that we are talking about a
Roman name that is so ancient that it might actually be Etruscan, "attendant at a sacrifice" and "attendant at a religious service" basically mean the same thing; most "religious services" well over two thousand years ago in what's now Italy would have involved some sort of sacrifice to the gods.
What sites tell you that
Camille or
Camilla goes back to "swift runner" or "innocent"? Most of the reliable reference books I have simply say "obsucure" or "unknown" for the original meaning of
Camille or
Camillus.