Although not used in the movie, it's very likely that the term jan was made to talk about the "Genesis device" in Star Trek III: The Search for Spock.
Regarding definition #2 – musical instrument:
In KGT it is explained that QoQ jan (musical instrument) can be shortened to simply jan. So basically, since a musical instrument is viewed as a device, it is not listed in this page as a second, different definition.