At the bottom of 108a, Rabbi Yeshoua Ben Levi says that women are obligated in the 4 cups. Rashi points out that it says the word cup (kos) 3 times in the verse in which Pharaoh's wine steward relates his dream to Joseph (end of P. Vayeshev). The 4th cup is for Birkat HaMazon. Why does Rashi give this explanation and not the standard explanation he used on the Mishna (that the 4 cups complement the 4 descriptions/adjectives/stages of redemption)?
The only possible answer I thought if is perhaps Rashi is worried about the Gemara about pairs -- and follows Rava's opinion that the cup used for Birkat Hamazon can't count against you. So although we have 4 cups, they are not pairs, as 3 (odd number) are for "kos" and one is for Birkat HaMazon. The standard explanation (4 leshonot of geula) would be the extension of the opinion that each cup is a separate Mitzva.
Your question is very interesting. Your proposal is good, but we would still need to understand why Rashi is following Rava's opinion here, when it is not discussed until 109b, and it is only one of several opinions. Also, why does Rashi earlier (99b, DH Arba Kosos) mention that there are four cups because of the four terms of redemption? The CHASAM SOFER (99b) makes note of this point but gives no answer, and the MAR'EH KOHEN (on the Yerushalmi) and TASHLUM YEFEH EINAYIM also ask the question.
(a) We find that Rashi's reason here for the four cups -- that they correspond to the cups of wine mentioned in the verse with regard to the butler's dream -- is mentioned in the Yerushalmi (beginning of Arvei Pesachim), and in the Midrash (Bereishis Rabah 88; Shemos Rabah, beginning of Va'era). It is one of several reasons given for why we drink four cups of wine on Pesach night. The reason of the four terms of redemption is also mentioned there. The Tashlum Yefeh Einayim suggests that Rebbi Yehoshua ben Levi, the author of the statement on 109b, is the one in the Yerushalmi who gives the dream of the butler as the source for four cups, so Rashi mentioned that reason over there. The Midrash, though, cites it from a different Amora.
(b) The SEDEI CHEMED (Ma'areches Chametz u'Matzah 15:10) quotes a letter from RAV CHAIM BERLIN, son of the Netziv, who deals with your question. He suggests that the reason of four Leshonos of Ge'ulah doesn't apply to women, since some of the forms of slavery represented by these four Leshonos didn't apply to women (only men, and not women, were drowned). That is why Rashi doesn't mention the four Leshonos on Daf 109a, where it is discussing the obligation of women to drink the four cups. On the other hand, if the fourth cup is Kos Shel Berachah, he suggests that it should not be supplied from the Tamchuy for a pauper, any more than any other Kos Shel Berachah. Therefore he mentioned the four Leshonos of Ge'ulah in the Mishnah. (This answer is somewhat forced, though, for a number of reasons, some of which the Sedei Chemed there points out.)
(c) However, the Yerushalmi and the Midrash mention that there were four cups of wine in the Parshah of Pharaoh's dreams (the fourth one being the cup which is mentioned in the interpretation of the dream, besides the three mentioned in the dream itself), and not three as Rashi says here. Rashi, then, is even more difficult to understand, for not only does he choose to give a different reason than the one he mentioned earlier, but he does not quote the reason properly as it is stated in the Midrash! Why does Rashi say that three cups correspond to those in the Parshah of Pharaoh's dreams, and the fourth one was instituted for Birkas ha'Mazon? (See also Chulin 92a, where the Gemara analyzes the three cups of wine in the Parshah of Pharaoh's dream, and does not discuss the fourth cup, mentioned in the interpretation of the dream. However, the fact that the Gemara there does not make a Derashah from the fourth cup is no reason to say that it does not serve as a source for the four cups of Pesach.)
It seems, therefore, that the explanation of Rashi here is as follows.
Rashi was bothered why the Gemara had to say that "women are obligated in the four cups." It is obvious that they are obligated! Why would we have thought that they are not obligated? Even though Tosfos (108b, DH she'Af) writes that we might have thought that they are exempt because it is a Mitzvah Aseh she'ha'Zman Garmah (even though this Mitzvah is mid'Rabanan, nevertheless the Rabanan make their enactments based on the guidelines of Mitzvos d'Oraisa), this approach is difficult because we know that women are obligated in all of the Mitzvos of Pesach night (Matzah, Hagadah, and even Korban Pesach according to some opinions, see 91b), and if so there is no reason why they should be exempt from the four cups! Why, then, does the Gemara have to tell us that they are obligated?
This is what Rashi wants to answer. Therefore, Rashi says that the Chidush of the Gemara is that women are obligated in four cups as opposed to three. We might have thought that they are not obligated to drink the cup of wine which is used for Birkas ha'Mazon, because we never find that a woman is obligated to use a cup for Birkas ha'Mazon. We might have thought that women are obligated to drink only three cups, excluding the one used for Birkas ha'Mazon. Therefore the Gemara teaches that they must also drink four cups, like the men, because they, too, were involved in the miracle.
In the Mishnah, though, Rashi gives the simpler reason for the obligation of four cups, that they are related to the four Leshonos of Ge'ulah (which is much more directly related to Yetzi'as Mitzrayim than the butler's dream), a reason which is mentioned explicitly in other Talmudic sources.
I hope this helps. Be well, Mordecai
Please note that the question posed by Prof. Schiffmiller was also asked by the Bigdei Yesha in his commentary on the Mordechai which appears in the back of the Vilna Shas on Daf 34b (p.68), note 7.
The Bigdei Yesha explains that Rashi felt that each reason on its own is inadequate. Based on the first reason, i.e., the four expressions of Geulah, one might have concluded that one should take 4 matzos instead of 4 cups. Therefore, I need the second reason, i.e., the three cups mentioned in the butler's dream. On the other hand, the second reason on its own is not enough because (i) the butler's dream has no connection to the Jews' redemption from Egypt and (ii) one might conclude that one only needs 3 cups altogether, including the Kois shel Brocho, not 4. Therefore, I need the first reason relating to the 4 expressions of Geulah.
Also see the Mordechai ad. loc. who gives another reason for the 4 cups.
Best regards and continued Hatzlocha.