Answered by Sayyidi Habib Umar bin Hafiz (may Allah protect him and benefit us by him)
Regarding the entrance of Ramadan and the day of Eid, if my family does not follow the moon sighting, should I follow the moon sighting or do what my family is doing to avoid conflict?
You must submit to the correct opinion according to the Sacred Law without disrespecting your family and without argumentation or animosity. If they celebrate Eid on a different day then express your joy and give them your good wishes even though you yourself are not celebrating it on that day.