Nice summary. I don’t see any reason for this. The Jews calculate Passover based on the old or new Julian calendar?
It doesn’t really make a difference exactly what day it’s on. I mean we celebrate Xmas the same time.
Toss a coin and go with that.
In the time of Moses, Passover was in the first month (see Exodus 12.2), but later the Jews adopted the Babylonian calendar in which the New Year is in the fall. Passover begins on the 14th day of the month in the evening which would be at full moon particularly if they begin the count the evening that the crescent is first visible after sunset. How it is done now I can't tell you, whether by observation or by calculation.