We still count, just as we always have, each month, from new moon to new moon.
Nisan 1 starts not with a visible event because the Great Sanhedrin is no longer in existence to validate it. It is calculated astronomically, as decided by the rabbinical patriarchate of the Great Sanhedrin.
Prior to the fall of the Temple and the later Bar Kokhba Revolt, the Great Sanhedrin was moved from Jerusalem to Yavne and then to Tiberias.
Because of this the Great Sanhedrin couldn't declare for the Diaspora when months officially began.
It was before the Great Sanhedrin dissolved that Hillel II, around the year 358 CE, created the current Jewish calendar. Hillel II introduced fixed standardizations that aligned via astronomical calculations, preserving the Jewish lunar year. He also introduced improvements allowing the addition of leap months to keep the calendar aligned with the solar year as well as allow for an automatic determination of exactly when the new moon appeared over Jerusalem, good or bad weather, something impossible before. (You had to do visible determinations prior to this, weather permitting.) You could also go backwards in time to precise dates in history using the Jewish calendar and Jewish dates, unlike before, as well as into the future.
This invention proved to be timely as the Great Sanhedrin was disbanded in 425 CE.
Counting 14 days from from Nisan 1 is always Nisan 14, that is correct. Again, it is not a visible system.
You cannot rely on the Spring Equinox due to leap years.