I was taught to pronounce it like English "w".
Am I wrong, are they wrong, are there different competing approaches to "correct" Latin pronunciation, or are the two sounds in free variation?
During the evolution of the language, the pronunciation of Latin "v" has changed to that of English "v", so this is how all Romance languages pronounce it.
Actually the sound "v" did not exist in the ancient Indo-European languages, but only the English "w".
The change in pronunciation from "w" to "v" has affected most European languages, not only those descended from Latin, unless they have lost completely the original "w", like Greek.
English is an exception where the ancient pronunciation of "w" has been conserved, even if other sounds have changed more than in most languages.
> How should one pronounce the cae- in caelum (sky): kai, as it was in Rome of the 1st century BCE, or che, as used in Christian liturgy and modern Italian?
but then later concludes when discussing a meeting of students held in Latin:
> The meeting removed, however, any concerns I had about sound and style in speaking Latin actively. Differences of pronunciation and expression mattered as little as if I were conversing in English with students from Scotland, Germany or Japan. What mattered was that the language was intelligible, meaningful, accurate and alive.
So in short, if people understand you, that's what matters, and the differences can be thought of as differing accents.
In areas where Catholicism has predominated, it's more likely that the ecclesiastical pronunciation will dominate. As I recall, the German exchange students I had in some of my classics courses used the classical pronounciation (their big weirdness was pronouncing θ as an aspirated t (like in Goethe) rather than a th sound.
Americans pronounce 'r' where British do not. Not a problem. There is no "correct" in a language which lived for centuries and does no more.