I would disagree about the computer x calculator thing, or more precisely about Babbage's machine begin computer instead of calculator. I don't think if-then is enough.

I think first actual computer emerged when Von Neumann architecture was born, when program and data became one. That was in theory when Turing machine was extended to do that (I don't know when), and in practice with SSEM (or Baby) built by Turing and co. in 1948, combining both no data - program distinction, and sufficiently strong computation capabilities.