Lapsed Advanced Ground Instructor certificate, can I take a FIRC and and restore my privileges?
First, as the regulation says, you can indeed take an FIRC to restore your privileges.

Second, I couldn't find any FAA clarification on exactly what "employment or activity as a ground instructor" means, but the language is fairly clear: you just have to be able to show that you were providing instruction as a ground instructor. That could be via training records, invoices, employment contracts, written test endorsements etc.

As a general rule, there's no need to look for extra requirements that aren't stated in the regulations.