There are differing customs regarding raising the cup during Hallel. The Beis Yosef (end of 422) seems to take it as a given that the cup is raised. The sefer Seder Haaruch suggests that the reason mentioned in your question is indeed the source for this custom.
On the other hand, the Aruch Hashulchan (473:23) writes that 'אצלינו' the custom is to put the cup down immediately after לפיכך, so apparently he doesn't agree regarding the reason of saying shirah on wine. He explains that the reason we raise the cup by וְהִיא שֶעָמְדָה and לפיכך is based on the verse כּוֹס יְשׁוּעוֹת אֶשָּׂא - which can be interpreted homiletically to mean that we raise the cup when we mention salvation. This idea is also found in the commentary of the Rokeach, who also implies that this applies specifically to national salvation.
I think perhaps we can explain your custom by extending this reasoning to cover the first Hallel as well, which prominently features the salvation from Egypt (the commentary attributed to the Rashbam understands even the first psalm as referring to yetzias mitzrayim). The second hallel is not about national salvation, but is rather more of a general praise, so we don't raise the cup.
So perhaps those who raise during the first Hallel for the reason mentioned by the Ta'amei HaMinhagim are indeed those who also raise during the second Hallel.