Actually, AFAIK the cups of wine really have no formal designations in Judaism. There is the Ashkenazic custom of filling the 'Cup of Elijah after the third cup (and the ‘Cup of Miriam’ that some also fill beside Elijah’s at this point, just to be gender-equal

), but beyond this, the cups are just called, for lack of a better term, ‘first cup’ or ‘second cup’ and so on. Now a number of folks today (mostly Christians or Messianic Jews) may try to apply different names for each cup, but these are all unofficial, and the names could vary depending on the Seder.
With all due respect to Mr. Hahn, I think that the one flaw in his interpretation (which I once held actually) is the assumption that Passover meals during the Second Temple period were performed like modern Seders. On the contrary, it is more likely that the Passover meal - like many other aspects of Judaism - was not yet fully developed in Jesus’ time and was yet still in a fluid, unfixed, evolving form. We can’t just assume that the ritual stayed as it was after more than 2000 years; we must reconstruct, and it is difficult, if not impossible, to reconstruct exactly what happened in a 1st century Passover celebration.
We do know the basic structure and the basic elements of a Passover meal, but what I’m saying is that it is hard to reenact in detail what Jews two millenia would have specifically done. It’s not like they had Haggadahs handy like today.

I almost expect that every household or every community would have had their own ‘flavor’ of celebrating the feast, doing things as their fathers and their fathers before them have done. While rabbinic works such as the Mishna and the Talmud does purport to transmit the way many things were done in those days when the Temple still stood, we can’t be too reliant on them and take them on face value as sources for Second Temple Judaism, since it seems that they often actually reflect more ‘how we (the rabbis) would have wanted things done’ than ‘how things were done’. In short, more of an ideal fantasy than historical reality.