The original methodolody of the Fixed Hebrew Calendar was for Rosh HaShanah to start at Sunset of the day which contains the Molad. It is clear from the Gemora, “You deliberately” (RH 25a), that if the Calendar Court should elect to use an earlier or later date then such would be valid. Such changes were called Dehiyyot. They are discussed in my book.
The purpose of the Dehiyyah Molad Zaqen seems to confound the calendar commentators. This dehiyyah defers any Molad in the last quarter of the day. Its purpose becomes quite clear from the Molad of Tishrei Study. This dehiyyah eliminates all 3-day discrepancies. 3-day discrepancies are synonymous with a very visible Old Moon Erev Rosh HaShanah.