This is incorrect. In a single-consent state, as long as YOU are a PART of the conversation (as Neal stated), you don't have to tell them anything. You are totally allowed to inform them that you are recording the call, but you have no obligation to pass along that info. California is a two consent state... so in CA, for example, if Neal had done the same thing, he would have been evicted... and when it "went to court", the judge would have told him that the recording was inadmissible because he didn't have the property manager's consent to record the conversation.
The reason that telemarketers and customer service people give that long spiel over the phone is because of states with a two party consent laws. When a manager is training someone in a call center, they tell them, "read this script to the person before you say ANYTHING to them....." and that is because they have no idea where they are calling 1/2 the time. Most call centers don't even "dial a number".... they are on headset speaking to someone, and after they end the call, a computer dials the next number automatically. They don't know if that number is Louisiana or California (a two party state).
OP
Don't try to hide the snake. It's much more of a hassle than you think, and not worth being evicted over later because you were afraid to ask (and get it in writing) before you moved in.