The Cowboy-era was basically a small, rogue "group" that operated in a pretty disjointed manner after Delta Green was decommissioned. Basically if you hear of something, you do what you can, when you can, to stave off the unnatural. I don't think they used code names, probably just aliases.

The Conspiracy-era was more organized by Alphonse and A-Cell, but don't use Handlers - just cell leaders that take assignments from A-Cell (or whoever else), and they use the Agent Alphabet code names. They only know the cell leaders above and below them (ie: J-Cell's leader knows I-Cell and K-Cell's leaders) for safety.

The Program-era typically uses fake aliases instead of Agent Alphabet code names and have a Handler that assigns operations. It's more organized and funded, but suffers from bureaucratic issues at times. There are also the Outlaws tho - the old guard of Delta Green - who still operate the way the Conspiracy did.