Can I be claimed as a full-time student when I'm considered one by my school but the term I had did wasn't 5 months long.

Yes.

"A full-time student is a student who is enrolled for the number of hours or courses that the school considers to be full-time attendance.

To qualify as a student, the person must be, during some part of each of any five calendar months of the year:

1. A full-time student at a school that has a regular teaching staff, course of study, and a regularly enrolled student body at the school, or
2. A student taking a full-time, on-farm training course given by a school described in (1), or by a state, county, or local government agency.

The five calendar months do not have to be consecutive.

This information is found in the Personal Exemptions and Dependents chapter of Publication 17, Your Federal Income Tax."

If you were a full time student from Aug 31 (or earlier) to Dec 1 (or later) you meet the 5 month rule (parts of 5 calendar months). 

Meeting the 5 month rule only matters for whether your parents can claim you, as a dependent, under the Qualifying child rules.  You do not need 5 months to claim an education credit.