Which timing should be adopted for the Dhuhr prayer when there is a discrepancy in prayer times between two mosques in the same city: should I pray before the time of the mosque where I usually pray at home if the timing of the other mosque is safer, or should I wait until the time they have specified?

The fundamental principle for determining prayer times is based on the Shar'i (Islamic legal) signs, not written calendars. However, it is permissible to rely on trustworthy calendars. If you are able to determine the time for Dhuhr prayer using the Shar'i signs, then do so. Otherwise, do not pray until you are certain or highly confident that the time has entered.