The bi in binary means 2, so we use two digits which happen to be 0 and 1. (base 2)

A trinary system would be base 3 and yes would be the digits 0-2.

Commonly we use base 10 which is the digits 0-9

If you mean why we use binary and not trinary in computers, it's because a binary bit is easy to represent physically. Either it's on or off, something is there or it isn't, and it's very easy to detect/read by hardware