In most places in Israel (and some places in America) there are two times listed for Tzais Hakochavim. The earlier time is for the earliest Shema and the later one is for the time that Shabbos ends. In most places in America they did away with the earlier time in order to avoid confusion, but you are allowed to say Shema earlier than when Shabbos ends and it is considered Bizman. According to Rav Schachter, the earliest time to say Shema is Tzais according to Rav Schneuer Zalman of Liadi, which is 24 minutes as degrees after Shekiah.
Originally the Minyan Davened Heichi Kedusha, but they switched to a full Shemoneh Esrei a while ago. However, when the Chazzan used to finish Kedusha a number of people would leave and we would lose a Minyan of people answering for Chazaras Hashatz. Therefore, in order to make sure that we don't lose a Minyan, the 4:18 Minyan can do Heichi Kedusha, as per Rav Schachter and Aruch Hashulchan (OC 232:7)
In the past we got our Zmanim from MyZmanim, but we decided to switch to a different calculation that we believe is more accurate. Whereas MyZmanim calculates based on the zip code (10033) and does not reveal their formula for calculations, we calculate based on the exact latitude and longitude of Yeshiva University and use a formula that was developed by the National Oceanic and Atmospheric Administration and is open source; this amounts to approximately a 3 second difference. See this article from KosherJava which goes into more detail.
Because Rav Schachter thought that would be a fair compromise between having Kedushat Beit HaKensset and minimizing disturbing those who wish to continue Kollel Seder on the first floor.