I started learn to read the Torah some 75 years ago in Flatbush and was taught to read both ways. We did not repeat the entire pasuk but only the phrase "timcheh et zeikher Amalek". I don't remember if I was taught the order but, many years later, I decided to read the incorrect zekher first and then correct myself by saying zeikher the second time. Similarly, in the megilla, when the megilla has the incorrect bifneihem and laharog I read as written and then correct myself by repeating the phrase only: v'ish.... lifneihem and k'hashmid.... v'laharog. If the megilla is correct I read only once.
Anyone slightly interested in the subject should read R'Mordekhai Breuer's article. Those very interested should go directly to R' Prof Penkower's article. He gives the entire history, about 45 pages, including statistics on manuscripts from the time of ben Asher as well as customs of reading, etc.
The double reading evidently started slightly before the Hafetz Hayyim made it popular. Penkower cites a ba'al Kriah who was instructed by R' Sh'neur Zalman mi'Lublin to read twice. This R' Sh"Z died in 1902. The Mishna B'rura was printed in 1906.
I was delighted to read that R' Penkower read twice despite his indisputable proofs that zeikher is correct. A few years ago I gave a Friday night talk proving that there is absolutely no justification for double reading. On the following morning I read parashat zakhor and read zekher followed by zeikher. I too do as I was taught and do not allow myself to be confused with facts. I am overjoyed to find that I follow the derekh of the expert.
It should be pointed out that no eidah other than the Ashkenazi ever questioned the correct reading. AFAIK, Yekkes are the only Ashkenazi unaffected by the double-reading syndrome.
As to developments in Israel: One of my grandchildren told me his rosh yeshiva told him to read only once (based on Breuer). On the other hand I davenned this year at a yeshiva where in addition to reading in Abazit, Ashkenoz, Moroccan, and Yemenite they also had Parsi and what might have been Iraqi.
I have also heard a ba'al k'riah reading in mivta Ashkenazi repeat Machlas and Mochlas, and yahalom and yohalom. He told me that he was told to do so by Harav Nebenzahl. At this rate, it won't be long before we'll be hearing hundreds of p'sukim read twice.