When Paroh pleads with Moshe to pray to Hashem that He should stop the plague of ברד, hail, Moshe says that when he leaves the city, he will spread out his hands to Hashem, and the plague will indeed end (שמות ט:כ"ט). The Pardes Yosef on this Posuk takes note of the fact that Moshe never says that he will actually ask Hashem to stop the hail, just that he will stretch out his hands. When he does approach Hashem (שם פסוק ל"ג), he indeed never specifically requests that the hail and the rain stop, he merely stretches out his hands and the plague ends. The Pardes Yosef explains that although Moshe stretched out his hands in prayer, he did not want to ask that there be no more rain or hail, because rain, of course, is generally necessary and beneficial, and it is improper to request that something good should stop, even if there is too much of it. Moshe therefore simply stretched out his hands, trusting Hashem to respond appropriately to his silent prayer.
This idea that one shouldn't ask Hashem to take away something good is found in a Mishnah in Taanis (דף י"ט.) which describes different communal difficulties and tragedies because of which the Chachamim would declare public fast days, featuring special Tefillos. The Mishnah states that they would never declare such a fast day if there was too much rain. The Gemara there (דף כ"ב:) explains that this is because it is inappropriate to daven to Hashem because one has too much of a good thing. The Gemara adds, however, that if the excessive rain is actually damaging or dangerous, then one could daven for it to stop because it is then no longer a good thing. The same Mishnah then relates the famous story of Choni Hame'agel who "persuaded" Hashem to cause the rains to fall by "threatening" to remain standing in the circle he drew on the ground until it would rain, which it then did, coming down at one point with such force that Choni had to request that it fall in the proper measure. The Gemara (שם דף כ"ג.) elaborates on this story, saying that when the rain fell very hard, Choni's students asked him to daven that it should stop entirely, to which he responded that he has it by tradition that one doesn't daven because of having too much good. He solved the problem using only specific and precise language and actions. The Rambam (פרק ב' מהל' תעניות הלכה ט"ו) and the Shulchan Aruch (אורח חיים סימן תקע"ו סעיף י"א וסימן תקע"ז סעיף א') rule in accordance with all of the above.
This issue of not asking Hashem to hold back something good even when we don't need it is relevant to us today concerning our practice regarding the last recitation of Mashiv HaRuach U'Morid HaGeshem in the spring. Based on the Mishnah at the beginning of Taanis (דף ב.), the Shulchan Aruch (או"ח סימן קי"ד סעיף א') writes that we start reciting this phrase in the second Beracha of the Shemoneh Esrei at Mussaf on Shemini Atzeres. The Yerushalmi in Taanis (פרק א', הלכה א', דף א:), discussing why we wait until Mussaf, quotes a view that an individual should not begin to say it until he has heard it from the Sheliach Tzibbur. The Shulchan Aruch (שם סעיף ב') accepts this view; the Ramo (שם) adds that prior to the silent Shemoneh Esrei of Mussaf, an announcement is made to begin saying Mashiv HaRuach U'Morid HaGeshem so that the Tzibbur will say it then too.
This same Mishnah in Taanis then indicates that we stop saying Mashiv HaRuach at Mussaf on the first day of Pesach, and the Shulchan Aruch (שם סעיף א') rules accordingly. Here, however, no mention is made of any announcements not to say Mashiv HaRuach any longer. In fact, the Ramo (שם סעיף ג') writes that the Tzibbur does indeed say it in the silent Shemoneh Esrei of Mussaf; only the Chazzan omits it in the Chazaras HaShatz, and the Tzibbur, hearing the Chazzan's omission, then leaves it out starting with Minchah. The Taz (שם ס"ק ט') explains that this is because any announcement not to say this phrase would be potentially confusing. The Magen Avraham (שם ס"ק ח') and the Be'er Heitev (שם ס"ק ה'), however, explain that this is really because an announcement not to daven for rain would be like davening that Hashem should hold back something which is generally a Beracha, and this is inappropriate. Elsewhere, the Magen Avraham (שם סימן תפ"ח ס"ק ד') actually quotes the aforementioned Gemara in Taanis which says that one shouldn't daven to Hashem because one has too much rain as the source for this practice not to announce that people should stop saying Mashiv HaRuach U'Morid HaGeshem.
This idea may relate to an interesting question. Why do we wait on the first day of Pesach until Mussaf to stop saying it? Why not stop already the night before, at Maariv? The Yerushalmi cited above discusses this question regarding when to start saying Mashiv HaRuach on Shemini Atzeres, and explains that we don't start at Maariv because not everybody is in Shul then. The Rosh in Taanis פרק א' סימן ב'() elaborates, saying that since people often stay home at night, those in Shul will therefore know to say it, those at home will not, and as a result, different people will be doing different things, which is inconsistent. The Rosh says that this reason actually explains specifically why we don't stop saying Mashiv HaRuach at Maariv; the Raavad (השגות הראב"ד על בעל המאור לתענית, דף א: בדפי הרי"ף אות ב') says even more clearly that this reason applies only to the first night of Pesach. In truth, however, this explanation is not needed. If no announcement is made to stop saying Mashiv HaRuach, and the Tzibbur must first hear the Chazzan's omission of it before they omit it, obviously it will be said at Maariv.
Regarding changing at Shacharis, the same Yerushalmi presents two reasons for not starting Mashiv HaRuach then on Shemini Atzeres. First, people who were not in Shul the night before may think, upon hearing it at Shacharis, that it was to have been said at Maariv too, and will make a mistake in subsequent years. Second, since an announcement to recite it must precede the Shemoneh Esrei, at Shacharis no announcement is possible, because there can be no interruption at all between the Beracha of Ga'al Yisrael and the start of the Shemoneh Esrei. We thus wait until Mussaf. Again, if no announcement is made to stop reciting Mashiv HaRuach anyway, it would be possible to stop at Shacharis on the first day of Pesach, at least during the Chazaras HaShatz, except for the Yerushalmi's first reason about causing confusion in subsequent years. The Aruch HaShulchan (סימן קי"ד סעיף ד') adds that the Halachos should be consistent; if we start saying Mashiv HaRuach at Mussaf, we should stop saying it at Mussaf as well, especially since there are more people in Shul at Mussaf time.
It is worth noting that according to the Minhag of those who daven Nusach Sefard (and many who daven Nusach Ashkenaz), Morid HaTal is recited in place of Mashiv HaRuach U'Morid HaGeshem. That begins at Mussaf on the first day of Pesach, and for that an announcement should be made, as the Mishnah Berurah (שם ס"ק ג') and the Aruch HaShulchan (שם סעיף ז') write. All the same explanations would then hold true for both insertions, as the Taz (שם ס"ק ג') implies. The Kaf HaChaim (שם ס"ק י"ד) writes that the special Piyuttim recited by some before the Shemoneh Esrei (or perhaps, we may add, the special tune used by the Chazzan for Kaddish) may constitute a sufficient announcement for the Tzibbur to begin saying the new insertion.