In the 'Amidah prayer on all Holidays, the name of the Festival must be mentioned. There is a difference of opinion as to how one needs to refer to Shemini 'Asereth. The Shulhan 'Arukh says that we must say את שמיני חג העצרת הזה (Eth Shemini Hagh Ha'asereth HaZeh). The Rama says that in Ashkenazi lands the custom is not to mention the word חג (Hagh - festival), because nowhere is it refered to as a festival. Instead, says the Rama, one should say, יום שמיני עצרת (Yom Shemini 'Asereth).
The Mishnah Berura quotes different opinions for Ashkenazim. He refers to the Peri Maghadim who says that one should say עצרת ('Asereth) without the letter Heh (ה) at the beginning, and also quotes the opinion of the Rashal and the Taz who write that one should say שמיני עצרת חג הזה (Shemini 'Asereth Hagh HaZeh).
The Ben Ish Hai's opinion is that one should say את יום חג שמיני עשרת הזה (Eth Yom Hagh Shemini 'Asereth HaZeh). While there are multiple opinions in this matter, as in all cases, each one should do whatever his custom is.
(See Shulhan 'Arukh with Rama 668:1. Mishnah Berurah, ibid., 3. Ben Ish Hai, 1st year, Wezoth HaBerakha, Oth 12)
Smini Atzeres, Shemini Atzeret, Shemini Asereth, Chag, Chagim, Hag, Hagim.