According to Shulchan Aruch, (please cite Saeef and Siman so I can read further):
When do we switch from Morid Hatal to Morid Hageshem and vice versa? What about Barechenu and Barech Alenu?
What happens if we say the wrong beracha?
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The Shulchan Aruch (Orach Chayim 114:1) states that "We begin saying (in the second beracha) Mashiv HaRuach U'Morid HaGeshem during the Mussaf prayer of the final Yom Tov [of Sukkot (i.e. Shemini Atzeret)] and do not cease until the Mussaf prayer of the first day of Pesach":
מתחילין לומר בברכה שניה משיב הרוח ומוריד הגשם בתפלת מוסף של יום טוב האחרון של חג ואין פוסקין עד תפלת מוסף של יום טוב הראשון של פסח
Here, too, is the Mishnah Berurah digest for se'ifim 1 and 2. The siman goes further and describes what to do in different situations (e.g. if you have made a mistake but have not yet finished the beracha, if you have made a mistake and have already finished the beracha, etc.).
This OU source suggests that some do not say it until Mincha, which makes a lot more sense to me since you are still obligated l'shev b'sukkah until the end of the day on Shmeni Atzeret.... https://www.ou.org/holidays/shmini-atzeret/jewish_prayer_for_wind_and_rain/
But honestly, I don't get why you would say it even at mincha, since you could be davening mincha in the early afternoon and still have several hours of d'orita sukkah sitting ahead of you before sunset on shmeni. Why would we start praying (or thinking about) rain until ma'ariv after shemini atzeret?
We joked that it takes time to process the paperwork... but seriously folks, why?