Personally, I view quotation marks as marks that surround a quotation. As such, anything that isn't part of the quotation shouldn't be within the marks. In other words, if I'm quoting @medica's comment directly, I would say, "I don't see that particular style of punctuation (AmE)." The period remains inside the quotation marks because the quote itself has a period in that position. If I were, however, quoting that (s)he said the word "particular", I would put the comma outside of the quotation marks, as the comma isn't part of what I'm quoting.
The same applies to the second example you provide. If I'm listing a series of nicknames, for example, I might tell you that growing up, my dad used to call me "Tiger", "Champ", or sometimes "Buckaroo". The punctuations are not part of the names he used to call me, so the punctuations shouldn't go inside the quotes.
I do believe, though, that your third example should have the comma inside the quotes. "That," she said, "is an abomination." My reasoning here is that I believe the character would most likely be pausing in her speech, thereby meriting a comma to show the pause. If she uttered the whole sentence without a pause, then the "she said" wouldn't be there to interrupt, at least in my eyes.
Of course, this isn't what I was taught in school at all. I was taught that the punctuations should always go inside the quotation marks because it looks neater, as @CotyJohnathanSaxman's link says:
According to Rosemary Feal, executive director of the MLA, it was instituted in the early days of the Republic in order "to improve the appearance of the text. A comma or period that follows a closing quotation mark appears to hang off by itself and creates a gap in the line (since the space over the mark combines with the following word space)."
But to me, it just doesn't feel right or make much sense.
Is the logical system of punctuation becoming more prevalent in the US with regard to the placement of periods and commas outside the quote marks?
To be honest, I'm not sure. I see quite a mix online, but I couldn't tell you which things I read are US-English or otherwise. I also couldn't tell you which ones of those things are "casual" posts -- I see a lot of "lol" and "gtg" online too, but that doesn't mean that internet acronyms are becoming more prevalent in serious writing.
I can tell you, though, that they still teach the "looks neater" version in US public schools. So if the "logical" version is becoming increasingly popular, it's probably due to its own merit.
(Thank you for this question, by the way, and for your link, @CotyJohnathanSaxman. Prior to this, I've always been nervous that I was wrong or looked stupid when I quoted things that way. I didn't know it was so common.)