Rosh Hashana seems to be a bit ambiguous. On one hand, there is the Tosphot in Sanhedrin 10b where the first opinion is the conjunction [molad] is the day of Rosh Hashanah. But then, most of that Tosphot points out the gemara in tractate Rosh Hashana where you find the time the Sanhedrin would set it was when the new moon was visible which is almost always a day later. And what makes this confusing is the fact that in Sanhedrin 10b you find that the time of Rosh Chodesh [new moon] depends on heaven. It says, ''If the earthly court sanctifies the new moon on time, then fine. But if not, the the heavenly court sanctifies it anyway''.
The calendar was adopted from the ancient Greeks. It is the Meton calendar and has little validity except in so far as when Jews were dispersed, no one had a set court that would tell them when the new moon was. At least during the time of the gemara, there was some stability. But at some point there was a revolution in Iraq and the yeshivot were closed for a hundred years. So at some point, Saadia Geon decided to accept the Meton calendar which more or less corresponded to when the Sanhedrin would have decreed Rosh Hashanah.
[The idea that Hillel the second set the calendar is a fiction. If that had been so, the gemara would have mentioned it.
My opinion is that it is best to go by the molad since there is no Sanhedrin nor any valid semicha. [I mean to say that at least if the was valid ordination, then any court of three judges with valid ordination could sanctify the new moon. But valid ordination disappeared in the beginning of the Talmudic period. [Some of the first Amoraim had ordination like R. Yochanan. But after that, it was lost. Ordination since then has been a fiction. ] [However, the ability to decide halacha was not lost. That ability continued until Ravina and Rav Ashi-as the Gemara says ''רבינא ורב אשי סוף הוראה.'' [''Ravina and Rav Ashi are the end of the ability to render a halachic judgment'']. Later, when the Geonim and Rishonim decide a halacha, they are trying to decide what the gemara holds.
Also I have to add here that I think the gemaras in Sanhedrin and in Rosh Hashana are in conflict. While in most cases Tosphot is right to reconcile gemaras that seem to be at odds, but this is one case that I do not think it possible.
