How to invoke osascript 'Generic Scripting System'?
According to OS X system internals, the "Generic Scripting System" is a generic class to contain the two flavors of scripting supported:


AppleScript
JavaScript


It's not some third option that adds one or more new dialects, but a container to hold the existing ones. This is indirectly explained in Apple's documentation - they list the two languages and don't mention the wrapper class (or whatever it's proper name is):


Mac Automation Scripting Guide -  part of which is the list of languages


If you needed other languages, I would call AppleScript "do shell script..." to then call whatever shell executable that ships by Apple or you have installed (bash, sh, python, swift, perl, ...).