My response depends on where I am parked. One late night in rural Virginia I parked off a gravel road up in a wooded area. After the cop informed me at 3 am that the land I was parked on was posted and I was trespassing, I apologized and prepared to leave. The cop then directed me to a truck stop a few miles down the road where I would be parked, then drove off. No demand for ID or request to search my truck.
Every other time I have been caught (sleeping or awake) the cops have told me that I am not breaking any laws, but that they just wanted to make sure everything is all right, or they got a call. They often ask for ID but I politely refuse.
I would never advise lying to a cop, or telling a half-true story. Most cops are astute at picking out liars and you may find yourself the subject of an interrogation which forces you to tell more lies and half-truths to back up your original story.
Personally, if a cop is being friendly and conversational I don't have a problem telling them about my travels. If they are playing hard-ass interrogator I say as little as possible.