If one is studying in a area about 500 miles from home, where he resides during term time, yet returns home once or twice a month, does he consider himself resident or traveler? Please mention full details including the opinions of different Mazhabs.

Praise be to Allah, the Lord of the World; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

Dear inquirer!
Know that the distance you mentioned in your question (500 miles) is more that the distance for which shortening prayers is required (80 Km). So, there is no objection to shorten your prayers. For details on this point consult our Fatwa: 83821.
Furthermore, know that you are not allowed to shorten prayers when you arrive to your country, since you become a resident not a traveler. One is not required to make intention in his place of residence to perform his prayers in full since this is the original rule that does need to be specified.
In addition, you are not allowed to shorten prayers in the place where you study as long as you remain in for more than four days.
Allah knows best.