On all Yomim Tovim we say "Chag Hassukos Hazeh" or "Chag Hamatzos Hazeh" but on Shmini Atzeres/Simchas Torah we say "Shmini Atzeres Hachag Hazeh" going in a different order. Why do we change the order specifically on Shmini Atzeres?
Actually, there are two different formats, depending on your nusach, but both different from the "standard":
- Shmini Atzeret Hachag Hazeh
- Shmini Chag HaAtzeret Hazeh
Either way, you should realize that the chag's name is not "Shmini Atzeret", but it is "an Atzeret on the eighth day".
That is, it's really an "anonymous" chag, and not part of Sukkot, but an "appendix" to Sukkot, and it should be on the eighth day.
This now explains both formats above: we actually say "... B'yom hashmini atzeret hachag" i.e. the eighth day which is an "Atzeret" for "The Chag" (i.e. Sukkot), or "chag haatzeret" i.e. it is a chag, but it is only known as "Atzeret".