According to the article it's OK for the cops to lie about almost anything. There's not a lot to stop them saying "I have the authority to grant immunity" or "I am an official from the DA's office" or "I just called the DA and he approved the immunity" or "I just called the DA and he approved the immunity and he sent this confirmation by fax".

Some of these might not really be de jure OK for the cops to tell you, but would you want to rely on a court upholding that?

They don't, so the absolute best case outcome of "asking a police officer for immunity" is "nothing happens".

The worst case outcome is that police officer testifying to a jury that you agreed to answer questions about a crime once you had immunity.

This is a terrible plan that ends terribly for you. Smart people think they'll be able to logic and reason themselves out of the crosshairs of the criminal justice system. But they're playing a game they don't know how to play and the odds are stacked for the house.

That's true. The cops would have to coordinate with the DA to make that happen. Which they would never bother to do unless they really needed your testimony, which is why it's a good way to get them to buzz off.