The Da'as Zekeinim you cite got this idea from Yerushalmi Pesachim 10:1. The Aruch HaShulchan, in his commentary on the Haggadah Leil Shimurim, addresses this question, of what do the cups in the story with the שר המשקים (the butler, as you call him) have to do with the 4 cups of the Seder. It's in his piece explaining why the four cups aren't mentioned in the Ma Nishtanah. The following is an excerpt from an English writeup of that piece.
The prophet Yirmiyahu describes what happened when the Jews were exiled from Israel by the nation of Babylon 1. He describes a vision of the matriarch Rachel crying for her children. Her cries echoed in the heavens and were accepted by Hashem. Why is it specifically Rachel? Undoubtedly the other foremothers were crying 2, yet only Rachel’s cries are mentioned in the verse.
The Jewish nation’s cries against the attacks from the Babylonians, the Edomites and the Amalekites stems from their overt baseless hatred they have for the Jews 3. Most nations when they conquer a different nation don’t destroy all of their buildings. Rather, once captured, they add the buildings to their nation. This is not true with these nations. When the Edomites captured Jerusalem, they didn’t stop there; they destroyed the temple and all the surrounding buildings 4. All of this came from pure hatred.
When Reuven goes before a King and pleads for help from what Shimon did to him, perforce Reuven can’t have done the same injustice himself. Otherwise, why should the King bother to help Reuven when he himself is guilty of the same crime? The same is true with the other foremothers, who were surely crying when their descendants were sent into exile. Their cries couldn’t have been answered, because their children were just as guilty of baseless hatred. The ten sons of Yaakov sold Yosef to the Egyptians due to their baseless hatred 5. Only Rachel and her children were clean from this sin. Yosef was the victim and Binyomin was too young to be involved in the sale. Therefore, only her cry regarding the baseless hatred being done to the Jews was answered 6.
The ruling of Pharaoh to reinstate the שר המשקים, the chief of the cup-bearers, and to execute the שר האופים, the chief of the bakers, seems to be backwards. What was their original crime? The שר המשקים failed to prevent a fly from entering the cup of the Egyptian King, and the שר האופים failed to notice a tiny needle which had fallen into the King’s bread 7. The crime of the שר המשקים is substantial; he literally handed the King a cup of wine with a fly in it! It’s not normal for the baker to hand the bread directly to the King. Even though there ended up being a needle in it, it’s not as much of an affront to the King’s honor. In fact, the bread might not have even reached the King’s plate. Why was the one with the smaller crime punished with death, and the other reinstated? Also, why such an extreme ruling? Pharaoh could have simply fired them both. Why was this considered a capital crime? The Torah even calls their crime a sin; it’s surprising to call simple lack of attention a sin.
A possible explanation is what was going on behind the scenes. The two ministers, due to their baseless hatred for one another, were brazen enough to try to get the other in trouble. The שר המשקים put a needle in the bread, and the שר האופים put the fly in the King’s cup. From this we see the sin of the שר האופים was much greater; he caused the שר המשקים to give the King a cup with a fly in it! That’s why he was sentenced to death. This is unlike the שר המשקים. Although putting the needle in the bread was a crime, there was a chance the King would have never even had that piece of bread. In the end, he was reinstated.
Another explanation brought for the four cups of wine is they correspond to the four exiles that were destined to fall upon the Jews. The meaning behind this is clear. It makes sense to drink a cup of wine as celebration for every salvation that is given to us, as the verse states 8: כוס ישועות אשא, the cup of salvations I will raise [to drink]. Since the four exiles will have four redemptions, we drink four cups of wine.
It’s clear that the reason for most of the exiles is the sin of baseless hatred 9. This is what the story of Yosef in prison is alluding to. Yosef was innocent from the sin of baseless hatred. The dream of the שר המשקים and its interpretation teaches us that just like baseless hatred caused them to be thrown in prison, the Jewish people’s baseless hatred of each other caused their four exiles. Each of these exiles deserves its own cup of wine to celebrate the salvation that was given to us, and the salvation that is yet to come.
1 Yirmiyahu 31:14
2 Cf. Eicha Rabbah Pesikta § 24 which only mentions the forefathers Avraham, Yitzchak, and Yaakov, as well as Moshe, crying and praying to Hashem alongside Rachel, with no mention of the other foremothers
3 See the prophets Amos 1:11, Ovadiah 1:1 and 1:12, and Psalms 137:7 who describe this baseless hatred
4 Even though the verse with Rachel crying is referring to the Babylonian exile, the Leil Shimurim seems to switch the focus of the discussion to Edom and its descendants, including the nation of Amalek
5 See Genesis Chapter 37. It’s not clear to me why it’s considered baseless hatred, the verses explicitly describe the source of their feelings towards Yosef
6 This also explains why King Shaul and Queen Esther, descendants of Binyamin, were specifically chosen to combat Amalek and no one from the other tribes. It also explains the concept of Moshiach ben Yosef paving the way for Moshiach ben Dovid
7 Rashi to Genesis 40:1, quoting Bereishis Rabbah 88:2
8 Psalms 116:13
9 Yoma 9b