The British way simply makes more sense. Indeed, since at least the 1960s a common designation for that style has been "logical punctuation." The best way to grasp this is to look at an example, such as what Slate commenter Dean Hamer wrote under a recent article about PBS and NPR:However, Yagoda notes: "When I asked [the Modern Language Association's Rosemary] Feal and Carol Saller, who oversees the Chicago Manual of Style, if there was a chance their organizations would go over to the other side, they both replied, in essence: 'How about never? Is never good for you?'"
[I]ronically, given the anecdote about "Tales of the City", PBS is the ONLY widely available channel that has any serious LGBT content; e.g. documentaries such as "Ask Not" and "Out in the Silence"."Tales of the City" and "Out in the Silence" are units—consisting of the words and the quotation marks. Insinuating a period or comma within the unit alters it in a rather underhanded manner. American style is inconsistent, moreover, because when it comes to other punctuation marks—semicolons, colons, exclamation points, question marks, dashes—we follow British/logical protocol. Dean Hamer would pass muster in any U.S. newspaper or magazine, for example, if he were to write: I am a big fan of "Tales of the City"; did anyone else see "Ask Not"?
...By far the biggest fount of logical punctuation today is Wikipedia, which was started by two Americans but whose English-language edition is by and for all English-speaking countries. The site's style guide notes that "logical punctuation … is used here because it is deemed by Wikipedia consensus to be more in keeping with the principle of minimal change." That is, if you put a period or comma inside quotation marks, you are wrongly suggesting that the period or comma is part of the quoted material, and thus you have "changed" it.
Survey says! By a 60%-40% margin, y'all are cool with the British way.