I'm British and, when I was at school, we were taught that punctuation should go inside the quotation mark. This was one of those horribly prescriptive rules, the breaking of which was considered very wrong indeed. I'm surprised that this article calls this "American Style", given my experiences.
I started ignoring the convention pretty when I started using computers because, as the article says, it's hard to defend it on merits and it just looks plain odd, especially in a fixed-width font.
I'm American, and I was taught to use the "British style". In fact, I didn't even realize there was an "American style"--I always considered it a punctuation error, though I guess it's common and inconspicuous enough that I didn't even realize publications like the NYT would make this 'error'.
I started ignoring the convention pretty when I started using computers because, as the article says, it's hard to defend it on merits and it just looks plain odd, especially in a fixed-width font.