You are not alone in objecting to the American convention. A whole group of kindred spirits is lurking on one of the links from the N.C. State Online Writing Lab homepage. The American system wins out, but the British system has surprising support. The analysis covers the reasons for the opinions. The rules generate a mess even worse than the one you supply.
All quotation marks fall outside terminal punctuation.
It has the character of a grammarian's party joke. After all, the punctuation is the least of its problems. Even orally, you can barely follow the words. The direct answer to your question is yes. In American English, the convention is to put commas and periods inside quotation marks.