The Talleli Oros Haggadah, beginning on page 21, lists several connections between the four cups and the four expressions:
I. R' Azariah Figo, Binah l'Itim §26: There were four aspects to the enslavement in Mitzraim:
- והוצאתי – that they made us poor through heavy taxes. Since they were instituted (in Shemos 1:12) למען ענסו בסבלותם, the passuk states that והוצאתי מתחת סבלות מצרים.
- והצלתי – that there was physical pain
- וגאלתי – that there was a loss of dignity in having been reduced to slaves
- ולקחתי – that we had a loss of the spiritual stature, and Hashem promises to take us as His people in response.
Based on these, we have four cups of wine:
- The cup on Kiddush, during which we say "Who chose us from all nations and sanctified us from all tongues," corresponds to ולקחתי.
- The cup on the Haggadah, during which we describe our freedom from slavery, corresponds to וגאלתי.
- The cup on Benching, during which we thank Hashem for providing all our needs, corresponds to והוצאתי.
- The cup on Hallel, during which we thank Hashem for freeing us from the physical pain of Mitzraim, corresponds to והצלתי.
(I'm not clear on how he connects והצלתי and וגאלתי to those aspects of the enslavement, nor on how he distinguishes that Hallel focuses on the physical pain; maybe someone with access to the original source can clear that up for me?)
II. Ma'aseh Nissim: In this version, the four languages reflect aspects of the leaving less so than aspects of the enslavement itself:
- והוצאתי – the miracle of physically leaving;
- והצלתי – the miracle of being redeemed from slavery, not just physically leaving but still being under their control;
- וגאלתי – Hashem taking us as His servants (cf. BM 47a which implies גאולה reflects acquisition)
- ולקחתי – Hashem taking us as uniquely His, comparable to a husband and wife (cf. Kiddushin 2a, that לקיחה implies marriage, and Kesuvos 71b, that our relationship with Hashem is comparable to that of a husband and wife)
The comparison with the four cups of wine, then, is:
- Kiddush, in which we declare Pesach the "time of our freedom," reflects והצלתי
- Haggadah, in which we describe actually leaving Mitzraim, reflects והוצאתי
- Benching, in which we describe Bris, Torah, and Eretz Yisrael, which all are physical representations of our servitude toward Hashem, reflects וגאלתי
- Hallel, in which we describe the ultimate redemption in the future, reflects ולקחתי; cf. Ibn Ezra to Shir HaShirim (no specific passuk noted) in which Yetzi'as Mitzraim is comparable to Kiddushin and the ultimate redemption is comparable to Nisu'in.