I think that this issue is important, but at the same time also difficult because these terms are not used consistently in programming. It would be interesting to see what other languages, eg Fortran, C, R, Python, and Matlab use for these concepts. This could provide some perspective.
I can help with R: it uses the term dimension more or less consistently for the dim attribute for an object, which is a vector of integers like Base.size in Julia. The R manual also calls this a dimension vector and dimensions, so usage is not super-strict. Docstrings (eg for apply, aperm) mostly use it in this sense and avoid other usages.