Is performing the "greeting of the mosque" prioritized over greeting those present in it, given that the greeting of the mosque is a right of Allah, and greeting is a right of the creation?

The first thing for someone entering the mosque is to begin with the prayer of greeting the mosque (two rak'ahs) before greeting those present. This is because the greeting of the mosque is a right due to Allah, and greeting people is a right due to them, and Allah's right takes precedence. This was the practice of the Prophet, peace and blessings be upon him, and his companions. It is sunnah for one entering the mosque to observe three greetings in order: Upon entering, to say: "In the name of Allah, and prayers be upon the Messenger of Allah," then to pray two rak'ahs as a greeting to the mosque, and then to greet the people. The greeting of the mosque is a recommended sunnah, not obligatory, according to the majority of scholars.