Many factions in wuxia are translated as Sects, either out of tradition or just habit, I'm not sure. But there are just too many and they don't really convey the meanings of what these factions stand for.
Here's the definition of the term sect:
A sect is a subgroup of a religious, political, or philosophical belief system, usually an offshoot of a larger group. Although the term was originally a classification for religious separated groups, it can now refer to any organization that breaks away from a larger one to follow a different set of rules and principles.
So it applies to groups like the Eastern/Western Sects of Wuliang Sword or the Sword or Qi Sects of Huashan Sword, since these are splinters of the original faction.
Whereas Wuliang Sword or Huashan Sword are factions that formed because people admired and wanted to practice the martial arts of the founder(s). This is usually because the founding master has a certain set of beliefs and principles in the style of martial arts. I'm still undecided on the naming for these, though it's easier to name factions that have very specific styles compared to those with broader beliefs.
Then there are those with religious backgrounds that are more of an Order, such as Shaolin and Wudang. Quanzhen would be considered a sect since it is a branch of Wudang.