The thing is, the "original" second temple wasn't very impressive. It was a downgrade from the first. It lacked the Ark of the Covenant, and several other holy articles that had been in the first temple. The Holy of Holies was separated only by a curtain rather than a wall. Most importantly, God wasn't present in the second temple in the way that he had been (according to the Biblical account) in the first.
This isn't just me bashing the second temple. The Hebrew Bible itself makes that clear in Ezra and Nehemiah. In particular, Ezra 3:12 describes how while everyone else was celebrating the temple being rebuilt, the people who were old enough to remember the original started weeping. While the original second temple is theologically important, it was basically an imperfect attempt to replace something which had been destroyed due to sin. Doesn't really make sense for a wonder. On the other hand, "Herod's" temple was a massive expansion, roughly doubling the area of the temple. The original building was replaced by a much grander structure. It wasn't technically a new temple, because they took care to make sure that rituals could continue during the construction work. But in terms of wonders of the world, it was effectively a new temple.
It's also very likely that if somehow the first temple hadn't been destroyed, Herod would still have built his expansion, and it would still be informally referred to as Herod's temple today. Even if the temple hadn't existed at all, or the second temple hadn't been built, he would probably have done some similar sort of project. The Temple was just a convenient thing for him to use as the basis for his monument he wanted to build.