I had a long discussion with my broker about this one. There's simply no logical reason why puts are classified as "covered" when there's a corresponding stock position with the same exposure. It seems as if somebody who meant "calls" wrote "options" in the regulations. The concept has no regulatory meaning, but I talk about "Cash-covered puts" when I sell puts and deposit the money to buy the stock in an interest-bearing account at the same time.
My broker is fond of selling deep out-of-the-money puts in parallel with a covered call position (I pay an annual fee rather than commissions). In case of disaster, the stock part of the covered call is sold at about the exercise price of the put (this makes me very nervous, but he does it with his own money as well as clients' money)