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":
Either way, you should realize that the chag's name is not "Shmini Atzeret", but it is "an Atzeret 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".