Every siddur, machzor, and bentcher that I've noticed writes בְּיוֹם, with a sheva, and then divides up into different cases (חג המצות/חג השבועות/חג הסוכות/השמיני חג העצרת, with additional options in bentching).
On Shemini Atzeret, it would seem that בַּיּוֹם would make more sense. On every other Yom Tov, and on Rosh Chodesh, יום is samuch to the following words, so it doesn't get a ה' הידיעה. But on Shemini Atzeret, my understanding is that we are giving two descriptions of the day, יום השמיני and חג העצרת, and יום is not samuch. I would therefore have expected בַּיּוֹם.
I checked the Artscroll Ashkenaz machzor, and it writes בְּיוֹם even in the Shemini Atzeret shemoneh esreis, when there's no need to divide into cases.
Does anyone know of someone who says you should say בַּיּוֹם, or any siddurim or machzorim that write it this way?
If not, what does בְּיוֹם השמיני mean?
(Note that I'm asking specifically according to those who say ביום השמיני חג העצרת, but feel free to include other minhagim in addition.)