How are volumes determined in manga? My question is about how are volumes determined in a manga. 
Does it go by major villain in a form of one per volume? Or... 
Does it go by story arc, in a way that a new story forms a new arc?
Lastly, are anime story series—arcs, seasons, sagas, etc—determined by the manga volumes?

Please help me understand how this categorization works.
There is no fixed rule to decide the boundary of a volume of a Manga. It's upto the mangaka and the publisher to decide where a given volume starts or ends. They're generally published arc-wise to maintain a uniformity throughout the volume.
As per Wikipedia

In Japan, manga are usually serialized in large manga magazines, often containing many stories, each presented in a single episode to be continued in the next issue. If the series is successful, collected chapters may be republished in tankōbon volumes, frequently but not exclusively, paperback books.
After a series has run for a while, publishers often collect the episodes together and print them in dedicated book-sized volumes, called tankōbon.