Toward the end of the Maggid section of the haggadah, we mention a dispute among the Tannaim as to how many plagues the Mitzrim suffered at the Yam Suf. What does this have to do with the mitzvah of retelling the story of yetzias Mitzrayim? In fact, the Rambam omits this section in his version of the haggadah. Rav Soloveitchik explained that this follows the Rambam's opinion (Hilchos Chametz U'Matzah 7:1) that on the night of the seder we are commanded to recount only the miracles that Klal Yisrael experienced in Mitzrayim and while leaving Mitzrayim, but not the miracles that occurred after yetzias Mitzrayim. Apparently, the author of the haggadah disagrees. He understands that even the miracles at the splitting of the sea are relevant to sippur yetzias Mitzrayim.
The Magen Avraham (67:1) takes this idea even further. He claims that one can fulfill the daily obligation to remember yetzias Mitzrayim by reciting the shiras hayam, the song that Klal Yisrael sang after the splitting of the sea. Both the Chasam Sofer and Rav Akiva Eiger (in their glosses to the Shulchan Aruch there) are troubled by this statement. After all, the possuk explicitly states that one is required to remember "the day that you left Mitzrayim (Devarim 16:3)" which implies that simply reciting the shiras hayam is insufficient. How can the Magen Avraham claim that just by mentioning the splitting of the sea one can fulfill the daily mitzvah of zechiras yetzias Mitzrayim?
The Talmud Yerushalmi (Pesachim 10:6) comments that although there is an obligation to sing shirah whenever Hakadosh Boruch Hu performs miracles for Klal Yisrael, nevertheless Klal Yisrael did not sing shirah when they left Mitzrayim because that was still only the beginning of their redemption. They did not experience a complete redemption until the splitting of the sea. Similarly, Rabbenu Bachya (Vaeira 6:6) writes that the phrase, "And I will redeem you with an outstretched arm and with great judgements" is a reference to kriyas Yam Suf since that is when Klal Yisrael achieved a complete redemption.
This idea is also hinted to in the fact that we do not recite a full Hallel nor do we say the bracha of shehechiyanu on the seventh day of Pesach. It is quite different from Shemini Atzeres, the last day of Sukkos, which is "a holiday of its own." (Sukkah 47a) Some explain that we do not recite a full hallel on the seventh day of Pesach because it would be inappropriate to sing a complete shirah for the splitting of the sea since that miracle also caused the drowning of the Mitzrim, and the possuk says, "When your enemy falls, do not rejoice (Mishlei 24:17)." But the Gemara (Arachin 10a) suggests a different reason why we do not recite a full Hallel on the last day of Pesach, and that is because it has no special korban musaf. Since its korban is the same as that of the first day of Pesach, it is not considered an independent yom tov, so it does not get a full Hallel of its own, and for the same reason we do not say the bracha of shehechiyanu. These halachos highlight the idea that the seventh day of Pesach, which commemorates the splitting of the sea, is not considered a separate celebration. But rather, it is viewed as the culmination of the celebration of yetzias Mitzrayim since kriyas Yam Suf was the time when Klal Yisrael achieved a full redemption.
What happened at the Yam Suf that made the redemption of Klal Yisrael complete? Rabbeinu Bachya explains that until the Mitzrim were drowned at the sea, Klal Yisrael were concerned that their former masters would chase after them and enslave them once again. But after the Mitzrim were eliminated at the Yam Suf, Klal Yisrael finally felt a complete sense of freedom since they no longer feared that they would be forced to return to Mitzrayim.
The Vilna Gaon (quoted in Kol Eliyahu, Parshas Bo) adds that the ge'ulah was not complete until kriyas Yam Suf when the Mitzrim were punished in the water, middah k'neged middah, in return for the evil they perpetrated against Klal Yisrael when they decreed that every Jewish newborn male child should be thrown into the river. The Netziv (Ha'amek Davar, Shemos 14:31) suggests that a similar idea is alluded to by the possuk, "And Klal Yisrael saw the great hand that Hashem used against Mitzrayim, and the people feared Hashem." This refers to how Hakadosh Boruch Hu meted out precise punishments for each and every Egyptian, corresponding to the pain and suffering that each one inflicted on the Jewish people in Mitzrayim. Rashi (Shemos 15:5) quotes the Midrash that the most wicked of the Egyptians were tossed around in the Yam Suf like straw, the average ones fell like stones, and the relatively decent ones sank immediately like lead. Each one received a punishment that was commensurate to his actions against Klal Yisrael.
After kriyas Yam Suf it says, "And they believed in Hashem and Moshe his servant. (Shemos 14:31)" Until then, the people could have deluded themselves into thinking that Moshe Rabbeinu had magically orchestrated the ten plagues and yetzias Mitzrayim. But when the people saw how precise the Divine punishment was, they had complete emunah in Hashem, and they realized that Moshe was only Hashem's agent in bringing about yetzias Mitzrayim.
That is why the redemption was incomplete until kriyas Yam Suf because one of the purposes of yetzias Mitzrayim was to instill in the hearts of Klal Yisrael a strong sense of emunah in the Ribbono Shel Olam. The ten plagues were designed to strengthen Klal Yisrael's belief in the existence of Hashem, Divine providence and omnipotence, and the concept of reward and punishment (see Maharal, Gur Aryeh, Vaeira 9:14). At the Yam Suf, this process reached its climax when Klal Yisrael saw the element of middah k'neged middah in the punishment of the Mitzrim. At that moment, they totally believed in Hashem and His power, and they appreciated His sense of justice. That was when Klal Yisrael achieved a complete redemption. (see Yarei'ach L'Moadim 67, for further elaboration)
It is no wonder that the author of our version of the haggadah includes the miracles of kriyas Yam Suf in the text of the haggadah, because it was only at the Yam Suf that one of the primary goals of yetzias Mitzrayim - namely developing a complete trust in the Ribbono Shel Olam - was finally achieved. This perhaps is also why the Magen Avraham rules that if one recites the shiras hayam he has fulfilled his daily obligation to remember yetzias Mitzrayim, because the miracles of kriyas Yam Suf, which are described in the shirah, were the catalyst that completed the process of yetzias Mitzrayim.