Unlike most Chinese dynasties that have seperate names like Eastern and Western Han or Northern and Southern Song, these two Jin dynasties have nothing to do with each other. One lasted from 265-420 after the 3 Kingdoms period, while the other lasted from 1115-1234 and is at its core a Manchurian dynasty which ended up invading the northern half of China.
The first one I mentioned I've only ever seen called the Jin Dynasty, while the second one I've seen called the Northern Jin, the Great Jin, and the Jurchen Jin (to my understanding, Jurchen was the previous term for Manchurian). Is Jin Dynasty a shorthand for one or both of them and they have official, distinguishing names, or is it just a difference one knows through context?
If you look at the Chinese characters they are completely different. The Jin from 265-420 is 晉 (fourth tone) while the Jin from 1115-1234 is 金 (first tone).