What was new about “The New Celebrity Apprentice”? “You’re fired” became “You’re terminated,” a catchphrase within a catchphrase — referring to Arnold Schwarzenegger’s career-making film, “Terminator” — that sounded clever for about two seconds. The setting moved from New York to Los Angeles, with a correspondingly sunny new opening sequence (still set to the O’Jays’ “For the Love of Money”).

And there was a new host, though if you hadn’t watched the 14 previous seasons of “Celebrity Apprentice” and the show from which it was spun off, “The Apprentice,” you might not have noticed. Donald J. Trump’s name wasn’t mentioned in Monday’s season premiere on NBC, appearing only during the closing credits, and the dreary two-hour episode sorely missed him. The closest the show came to acknowledging Mr. Trump was when Mr. Schwarzenegger, the new host, said: “I’m the new boss. And I intend to be tough but fair.”

Was that “fair” a dig at Mr. Trump, the businessman turned reality-television star turned polarizing president-elect, whose “Apprentice” tenure made him an international celebrity? NBC was in a bind, having claimed to sever ties with Mr. Trump after his first big campaign controversy (when he said of Mexican immigrants, “They’re rapists”) but still cutting him checks as an executive producer of the revamped show.

You can guess at the thinking in the boardroom. (The actual one at NBC, not the elaborate set where Mr. Schwarzenegger now terminates losing contestants.) The network needs to distance itself from Mr. Trump to protect its credibility, and after a strong start, the franchise’s ratings had been poor to middling for a broadcast-network reality series. But if the show appears to be running away from him, it might antagonize some of his constituency, not to mention displeasing a thin-skinned chief executive with influence at the Federal Communications Commission.