Should we shift the numbers and call category a 3-category?
No, but it seems to me that you are doing something very much like this.
Not at all. It may be was not a good example. A better example would be categories. If we follow the principle "foo = 1 foo" and want to agree with historical low dimensional terminology we should call categories 2-sets. Set = 1 Set. So, categories = 2 Set. Nobody will do it I guess. There are many other examples like stack, gerbes and so on. I agree with Mike Shulman that this is a byproduct of categorification. But we can survive with it. Concerning n-braided categories versus (n+1)-fold categories. Yes, I would be happy to use (n+1)-fold terminology but it also clashes with iterated monoidal categories of BFSW as I said. Michael. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]