taxonomy module -> classification module
I am fine with that.
Classification is fine.
vocabulary -> category group
Two words would be confusing, especially if category is used for term.
How about just "category" here.
Category is a problem because other systems (WordPress) use category to mean "term", and in many cases don't have a concept of multiple vocabularies or hierarchies. There needs to be a clear distinction between the vocabulary and the highest level of a taxonomy term hierarchy - that, I think, is the main issue at the moment. I'd go for something neutral like "container".
term -> category or tag
We can keep it as a term. The issue is the "vocab", not so much "term". Tag is more specific to free tagging, and may be overloading the term.
I think term is fine. Terms can be tags, tags aren't always terms - that can be made clear in help texts etc.