The difference is that Chertoff was being sincere - technically, if illegal aliens are working, then they ARE undocumented workers.
"Undocumented American" is clearly a parody. You'll notice no one has used the term since Dingy slipped up during a moment of weakness and expressed his true sentiments.
Throwing that term around at FR is no different than using a phrase like "let them eat cake" at a socialist website discussing the "plight of the people". It's either someone from the right tweaking the commies, or one of their own playing the pimp.
I suspect the latter - it looks like something I would do. In fact, it should enter the pantheon of other FReeperisms like hugh, series, etc. How's this: "We should ship all undocumented Americans back to their N/C/S American country of origin."
They ARE NOT undocumented workers. They are ILLEGAL ALIENS. Many of them have documents whether it is a phony SS card, a counterfeit or illegally obtained driver's license or the Mexican government issued 'Matricula Consular'
I am surprised that you are parsing such terms. I guess the Senate bill is, techncially, not amnesty. Give me a break.