When I invoke vi on a machine, I really have no idea what the implementation is. I imagine it is Vim on most Linux machines. I've been using vi for so long on so many types of systems that I only really use a safe subset anyway. I don't use any Vim extensions, so it makes sense to refer to it as vi as that is how I think of it. I imagine many others think of it the same way.

As far as I know, the original vi sees little use because it was caught up in the legal wrangling and fragmentation associated with the Unix Wars. However, some systems still have a clone that sticks closer to the original feature set. In particular, nvi was written by Keith Bostic specifically so that BSD would have an unencumbered vi implementation.

Probably because the person who invokes said editor war has little to no experience with vim.