The Roman Church uses (d) Latin and still
does in documents, etc. But the Latin it uses
with the C as in "ceiling": vici (vichi), is what I have
always called "ecclesiastical" Latin. Whereas the
other pronunciation of the "C" is like a K: vici, as in
Wiki. Is that the "classical" Latin to which you refer?

Well, the Latin that the Church uses is closer to Late Latin or Medieval Latin than to Classical Latin. The latter is what you find in Livy, Cicero, Caesar, and other classical authors. After Latin ceased to be a living, spoken language, its pronunciation started to change based on region: sort of like a regional accent. Ecclesiastical Latin tends to use the phonology of the people who spoke it the most, Italians. All the other major European nation states had their own pronunciation systems closer to the phonology of the underlying, native languages in each state. You can hear this in the older, legalese, Latin phrases of English jurisprudence.

The classical pronunciation used in teaching Latin these days is based on work that a bunch of philologists did in the 19th century. It is a kind of reconstructed classical pronunciation. If you want to read up on Latin pronunciation, you could try to have your library get a copy of W. Sidney Allen Vox Latina: A Guide to the Pronunciation of Classical Latin. There are older works that I have skimmed available for free on Google Books. If you want I can find some links later. Sturtevant wrote a Latin pronunciation book back in the earlier part of the previous century.

Ceci n'est pas un seing.