I'm not too keen on using that Wikipedia page as the source for the "usual definition" of smoothness for Frechet manifolds. There's a lot of details missing there. I usually quote the "Historical Remarks" at the end of the first chapter of Kriegl and Michor's book where they note that there are several inequivalent notions of what $C^\infty$ means even for Frechet spaces. That's sort of the point of their work: rather than seeing "smooth" as $C^\infty$ (ie the limit of $C^k$), they say "smooth is as smooth does" and start from what we would expect smooth maps to do. Explained right, this can actually be more intuitive and so easier to understand than "locally diffeomorphic to an open subset of a Frechet space".