Is Maghrib prayer considered offered outside its prescribed time if performed immediately after the Haram's Adhan, given the doubt about the actual entry of Maghrib's time, especially with the difference in Adhan times between the Haram and local mosques?

Prayer is not valid unless its time has begun. It is permissible to rely on the adhan of a trustworthy muezzin. If it is highly probable that the time has begun and the prayer is performed, then the prayer is valid and does not need to be repeated, unless it becomes clear that it was prayed before its time. Al-Buhuti said: "One acts upon the adhan of a trustworthy and knowledgeable person. If one begins the prayer based on ijtihad, meaning that the beginning of the time is highly probable in one's estimation... then their prayer is obligatory and does not need to be repeated, because the original state is the clearing of their responsibility."