It's a computer in any sense of the word. The only difference between the Analytical Engine and what you call an "actual computer" is speed. It had selection, iteration and sequence, it had working memory and it could load data and instuctions from punched cards.
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.