Let me begin by defining every other case of two Yamim Tovim (in Chutz la'Aretz). The reason that one observes the second day of Yom Tov in Chutz la'Aretz is due to Sfeika de'Yoma (a doubt on which of the two days possiblem Yom Tov was announced in Eretz Yisrael), and which was therefore a Safek in Bavel. As a result, they had to observe both days, although in reality, only one of them was really Yom-Tov.

Nowadays, we know that the first day is really Yom Tov and that one observes the second day because of Ma'aseh Avoseinu be'Yadeinu (in case Mashi'ach comes and we revert to the old system of witnesses and messengers). Consequently, the Rabanan allowed certain concessions (regarding the Dinim of burial and Refu'ah) on the second Yom-Tov.

Rosh Hashanah is different however, in that it applies even in Eretz Yisrael too. This is a clear indication that it was not instituted because of Sfeika de'Yoma (at least, not only because of Sfeika de'Yoma). But bear in mind the fact that they sometimes kept two days even in Eretz Yisrael, for entirely different reasons (as explained in the Mishnah to which you are obviously referring), and what's more, when they did keep two days, it was the second day that was really Yom Tov, and not the first.

Consequently, the Chachamim incorporated Rosh Hashanah in the Din of two days Yom Tov, only it is treated like one long day (Yoma Arichta [lasting forty-eight hours]), without the concessions that Sfeika de'Yoma permits.

May you and yours be Zocheh to a Kesivah va'Chasimah Tovah.

Kol Tuv

Eliezer Chrysler